on hurwitz multizeta functions - igm
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On Hurwitz Multizeta Functions
Olivier Bouillot1
Laboratoire d’informatique Gaspard-MongeCite Descartes, Bat. Copernic
5, boulevard DescartesChamps sur Marne
F-77454 Marne-la-Valle Cedex 2
Abstract
We give two new properties of Hurwitz multizeta functions. The first oneshows that each Hurwitz multizeta function is an example of a simple resur-gent function. The second property deals with the algebra Hmzfcv over Cspanned by the Hurwitz multizeta functions. It turns out that this algebrais a polynomial algebra with no other algebraic relations than those comingfrom the stuffle product, and thus is isomorphic to QSym.
Keywords: Hurwitz multizeta functions, Multizetas values, Resurgencetheory, Quasi-symmetric functions, Mould calculus.2010 MSC: 11M32, 11M35, 05E05, 33E20, 33E30.
Contents
1 Introduction 2
2 Elements of mould calculus 6
3 A fundamental lemma 11
4 Resurgence theory and Hurwitz multizeta functions 14
5 On the algebraic structure of Hmzvcv 41
6 Extension to the divergent Hurwitz multizeta functions 53
Email address: [email protected] ( Olivier Bouillot )URL: http://www-igm.univ-mlv.fr/∼bouillot/ ( Olivier Bouillot )
1Partially supported by A.N.R. project C.A.R.M.A. (A.N.R.-12-BS01-0017)
Preprint submitted to Elsevier March 6, 2015
1. Introduction
For any sequence (s1, · · · , sr) ∈ (N∗)r, of length r ∈ N∗ such that s1 ≥ 2,we define the Hurwitz multiple zeta function Hes1,··· ,sr+ over C− (−N), by
Hes1,··· ,sr+ : z 7−→∑
0<nr<···<n1
1
(n1 + z)s1 · · · (nr + z)sr. (1)
We also define He∅+ : z 7−→ 1 .The condition s1 ≥ 2 ensures the convergence of the series and the set
of sequences of positive integers satisfying this condition is denoted by S?:
S? = s ∈ N?1 ; s1 ≥ 2 , (2)
where N1 = N∗. The set of sequences of elements e ∈ E is denoted by E?.Let us also recall that a mould is a function defined over a free monoid,
or a subset of a free monoid like S? (See Section 2 for more details). Con-sequently, Equation (1) defines a mould, denoted by He•+, defined over S?and called the mould of Hurwitz multiple zeta functions (Hurwitz multizetafunctions for short).
These functions are a generalization of the classical Hurwitz function
ζ(s, z) =∑n>0
1
(n+ z)s, (3)
and, up to the knowlegde of the author, appeared in the literature at thebeginning of the XXIth century. The reader can find them under manyothers names, like “mono-center Hurwitz polyzeta” in [25], [42], “Hurwitztype of Euler-Zagier multiple zeta functions” (or also “Hurwitz-Lerch typeof Euler-Zagier multiple zeta functions”) in [27], [44], [45], and [46].
Today, these functions appear in different mathematical fields like:
1. the theory of special functions (see [24] for the particular case r = 2,[14] for a functional equation related to that of the classical Hurwitzfunction, [34] for the links between Hurwitz multizeta functions andthe Gamma function, [6] for the links with multitangent functions);
2. the multiple Dirichlet series, because they have an easy analytical con-tinuation in the variables si’s (see [1] when r = 2, [35], [41], [43] for awider class of functions);
3. holomorphic dynamics (see [4] and [6] for the links with analyticalinvariants);
4. the study of the Riemann Hypothesis (see [44], [45] for the localizationof the zeros of the Hurwitz multiple zeta functions);
5. renormalisation theory (see [27], [29], [40] for their values at negativeintegers and [8] for the link with Bernoulli polynomials);
2
6. quantum field theory (see [36]).
We also define He∅− : z 7−→ 1 and for any sequence s ∈ N?1 such that←−s = (sr, · · · , s1) ∈ S? the negative Hurwitz multizeta function Hes− by:
Hes1,··· ,sr− : z 7−→∑
nr<···<n1<0
1
(n1 + z)s1 · · · (nr + z)sr. (4)
1.1. Classical properties of Hurwitz multizeta functions
For any nonempty sequence s ∈ S?, the function Hes1,··· ,sr+ is holomor-phic over C− (−N∗), with derivative given by
d Hes1,··· ,sr+
dz= −
r∑k=1
skHes1,··· ,sk−1,sk+1,sk+1,··· ,sr+ . (5)
Such a function is also an evaluation of a monomial quasi-symmetricfunction Ms1,··· ,sr (See [28], or [3] for a more recent presentation). There-fore, this family spans an algebra Hmzfcv over C, where the product isthe product of QSym, known as the stuffle product (also called augmentedshuffle, contracting shuffle, quasi-shuffle, . . .: see [32], [22], [18], . . .).
More precisely, the stuffle, denoted by , is the product of the basis Mof monomial quasi-symmetric functions. It is recursively defined on words,and then extended by linearity to non-commutative polynomials or seriesover an alphabet Ω, which is assumed to have a commutative semi-groupstructure, denoted by + :
ε u = u ε = u .au bv = a (u bv) + b (ua v) + (a+b)(u v) .
(6)
Consequently, we have:
Hes1
+Hes2
+ =∑s∈S?〈s1 s2|s〉Hes+ . (7)
A mould which satisfies such a multiplication rules is said to be symmetrel .So, He•+, as well as He•−, is a symmetrel mould. With another vocabulary,we can say that He•+ and He•− are group-like elements of the Hopf algebrawhose product is the concatenation product and whose coproduct is the dualof the stuffle). See Section 2 for an introduction to mould terminology, aswell as [4], [6], [15], [20], [21] or [53].
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1.2. Connection with multizeta values
The family of Hurwitz multizeta functions is (at first sight) a functionalanalogue of multizeta values, which are defined by Zes1,··· ,sr = Hes1,··· ,sr+ (0)(See [12], [55] and [56] for surveys on mulutizeta values), that is, for allsequences s ∈ S?:
Zes1,··· ,sr =∑
0<nr<···<n1
1
n1s1 · · ·nrsr
. (8)
The story of these numbers goes back to 1775, to Euler’s article [26]which had first introduced these numbers in length 2 ( i.e. r = 2). His maingoal was to obtain an explicit formula for the evaluation at odd integersof the zeta Riemann function, in order to complete his computation foreven integers. Actually, he had been the first to discover the interest ofthese numbers, proving surprising relations such as Ze2,1 = Ze3, or moregenerally:
∀p ∈ N∗ ,p−1∑k=1
Zep+1−k,k = Zep+1 . (9)
These result have been forgotten during the XIXth century and for mostof the XXth century, but in the late 70’s, these numbers have been rein-troduced by Jean Ecalle in holomorphic dynamics under the name “moulezetaıque” (See [19], vol. 1, p. 137). He used them as auxiliary coefficients inorder to construct some geometrical and analytical objects. At first sight, hehad not seen all the algebraic interest of these numbers, even if he probablyknew their integral representation which is attributed to Kontsevich and isprecisely, today, the main point of their algebraic study.
But he has been the first one to show some connection between thesenumbers and another theory, namely resurgence theory (See [19] in general;for others connections with resurgence theory and analytical invariants, see[4] and [5], as well as Section 4 of this article, see also the introductorycourse [54] to resurgence theory). Other connections have then been discov-ered during the late 80’s (see p. 151 of [6]) and mathematicians have beendefinitely convinced of the interest of these numbers which eventually beganto be studied for themselves.
One of the questions about these numbers is the algebraic descriptionof their algebra over Q (denoted by Mzvcv). For instance, is Mzvcv agraded algebra? If yes, could we obtain a formula for the dimensions ofhomogeneous components? But, it turns out that answering questions likethat require arithmetical properties of multizeta values, difficult questions
1See Remark 4, p. 431 of the first volume of [19], where he refers the reader to Exercice12e4
4
on which only a little is known on this side, in spite of some recent research[9], [10].
Here, the question is a difficult one because multizeta values can bemultiplied in two different ways, using first the stuffle (which has been pre-viously defined), but also using another independent product of the stuffle,a shuffle product on binary words, coming from the Kontsevich’s integralrepresentation. Of course, the results of the two computations are numeri-cally the same, but have different expressions. Consequently, Mzvcv is analgebra with a rich algebraic structure. By making some shortcuts, JeanEcalle has shown at the begining of the XXIth century that, formally, thisalgebra is a polynomial algebra where the indeterminates are what he hascalled “irreducible multizeta values” (see [20] and [23]).
1.3. Results proven in this article
In this article, we prove two new results on Hurwitz multizeta functionswhich are consequences of an easy lemma. Hurwitz multiple functions satisfya 1-order difference equation:
Lemma 1. For all sequences s = (s1, · · · , sr) ∈ S?, we have:
Hes+(z − 1)−Hes+(z) = Hes1,··· ,sr−1+ (z) · 1
zsr. (10)
Section 2 will introduce these notations which will be used later on andSection 3 will deal with this Lemma.
The first result is part of resurgence theory, and can be stated in an easyway as
Theorem A. The Hurwitz multizeta function Hes+, with s ∈ S?, is a resur-
gent function. It is the same for the function Hes−, with s such that←−s =
(sr, · · · , s1) ∈ S?.
Section 4 is devoted to the proof of this theorem. It begins with a self-contained introduction to simple resurgent functions, as well as referencesof this subject and an easy, but quite useful example on a generic 1- orderdifference equation.
The second result has been announced in [7] and concerns the algebraicstructure of Hmzfcv. In general, elucidating the algebraic structure of analgebra, like Mzvcv or Hmzfcv, is a difficult question since it is often anequivalent question to the linear independence of elements of this algebra.For instance, linear relations could come from a second natural producthidden somewhere inside the algebraic structure itself, which is exactly thecase ofMzvcv. They could also come from another process (see the algebraof multitangent functions, for instance, see Section 4.4.1 and [6]). Even if we
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can imagine some similarities betweenMzvcv and Hmzfcv, the situation ofHurwitz multizeta functions is considerably simpler because we can prove thelinear independence of Hurwitz multizeta function over C(z). Consequently,the stuffle product, coming from the identification of the Hurwitz multizetafunctions with a specialisation of monomial quasi-symmetric functions, isactually the only product needing to be taken in account in Hmzfcv.
Summarizing this, the result can be stated as
Theorem B. The algebra Hmzfcv is a polynomial algebra, isomorphic to asubalgebra of the quasi-symmetric functions. Thus, all the algebraic relationsbetween Hurwitz multizeta functions, with coefficients in C, come from theexpansion of products, using the stuffle product.
Section 5 is devoted to the proof of this theorem. More precisely, we willprove first that Hurwitz multizeta functions are linearly independent overthe algebra of rational functions, and then deduce the theorem from thisfact.
In the last section, we introduce divergent Hurwitz multizeta functions,i.e. the functions Hes+, s ∈ N?1 with s1 = 1. Then, we extend the twoprevious results to these functions.
Finally, let us notice that there exists a lot of generalization of multizetavalues as well as of Hurwitz multizeta functions (already mentionned in theprevious citations). Let us mention another one, namely the colored Hurwitzmultizeta functions.
For any sequence(ε1, · · · , εrs1, · · · , sr
)∈ seq (Q/Z× N∗)− ∅, with the notation
ek = e−2iπεk , for k ∈ [[ 1 ; r ]], these are defined by:
He
(ε1, · · · , εrs1, · · · , sr
)+ (z) =
∑0<nr<···<n1
e1n1 · · · ernr
(n1 + z)s1 · · · (nr + z)sr. (11)
He
(ε1, · · · , εrs1, · · · , sr
)− (z) =
∑nr<···<n1<0
e1n1 · · · ernr
(n1 + z)s1 · · · (nr + z)sr. (12)
The two previous result can easily be extended to these functions, becausethe “colors”, i.e. the numerators e1
n1 · · · ernr , are complex numbers.
2. Elements of mould calculus
In this section we introduce briefly the notations of mould calculus.Many references can be found in many texts of Jean Ecalle. See for in-stance [20] or [21] ; other presentations of mould calculus are also available,see for instance [6], [15] or [53].
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2.1. Notion of mould
If Ω is a set, Ω? will denote in the sequel the set of (finite) sequence ofelements of Ω:
Ω? = ∅ ∪⋃r∈N∗
(ω1; · · · ;ωr) ∈ Ωr
.
A mould is a function defined over Ω?, seen as a monoid, or sometimesover a subset of Ω?, with values in an algebra A. Concretely, this means that“a mould is a function with a variable number of variables” (Jean Ecalle,from the preface of [15], for instance). Thus, moulds depend on sequencesw = (w1; · · · ;wr) of any length r, where the wi are elements of Ω.
Sometimes, it may be useful to see a mould as a collection of functions(f0, f1, f2, · · · ), where, for all nonnegative integers i, fi is a function definedon Ωi (and consequently, f0 is a constant function) .
In all this article, we will use the mould notations:
1. Sequences will always be written in bold and underlined, with an upperindexation if necessary. We call length of w and denote l(w) thenumber of elements of w and the empty sequence ( i.e. the sequenceof length 0) is denoted by ∅. We will reserve the letter r to indicatethe length of sequences.
2. For a given mould, we will prefer the notation Mωωω, that indicatesthe evaluation of the mould M• on the sequence ωωω of seq(Ω), to thefunctional notation that would have been M(ωωω) .
3. We shall use the notation M•A(Ω) to refer to the set of all mouldsconstructed over the alphabet Ω with values in the algebra A .
2.2. Mould operations
Moulds can be, among other operations, added, multiplied by a scalaras well as multiplied, composed, and so on. In this article, only the mul-tiplication needs to be defined: if (A•;B•) ∈ (M•A(Ω))2, then, the mouldmultiplication M• = A• ×B• is defined by
Mωωω =∑
(ωωω1; ωωω2)∈(Ω?)2
ωωω1·ωωω2=ωωω
Aωωω1Bωωω
2=
r∑i=0
Aω1,··· ,ωiBωi+1,··· ,ωr . (13)
Let us remark that the two deconcatenations ∅ ·ω and ω · ∅ occur in thedefinition and refer respectively to the index i = 0 and i = l(ω).
Finally, (M•A(Ω),+, ·,×) is an associative, but noncommutative A-algebra,with unit, whose invertible elements are easily characterised:
(M•A(Ω))× =M• ∈M•A(Ω) ; M∅ ∈ A×
. (14)
7
We will denote by (M•)×−1 the multiplicative inverse of a mould M•, whenit exists.
As an example, we have the following
Lemma 2. The multiplicative inverse(He•+
)×−1and
(He•−
)×−1of the moulds
He•+ and He•− are given on nonempty sequences s = (s1, · · · , sr) by:
(Hes1,··· ,sr+
)×−1(z) =
∑0<n1≤···≤nr
(−1)r
(n1 + z)s1 · · · (nr + z)sr, if s ∈ S? (15)
(Hes1,··· ,sr−
)×−1(z) =
∑n1≤···≤nr<0
(−1)r
(n1 + z)s1 · · · (nr + z)sr, if
←−s ∈ S? (16)
Proof. This is a direct consequence of the formula antipode for QSym (Seecorollary 2.3 of [39], p.973− 974)
2.3. Stuffle product
Let (Ω, +) be an alphabet with an additive semi-group structure. Let usfirst recall that the stuffle product (see [32]), denoted by , is recursivelydefined by:
ε u = u ε = u .au bv = a (u bv) + b (ua v) + (a+b)(u v) .
(17)
where ε is again the empty word.
Example 1. In N?, we have: 12 3 = 123 + 132 + 312 + 15 + 42 .
A visual representation of the stuffle product can be useful. Seeing aword as a deck of card, the stuffle of two words, a deck of blue cards anda deck of red cards, becomes the set of all the obtained results by insertingmagically one deck of blue cards in a deck of red cards. By magically, wemean that some new cards may appear: these new ones are hybrid cards,that is, one of their sides is blue while the other is red. Such a hybrid cardcan only be obtained when two cards of different colors are situated side byside in a classical shuffle. In the previous example, the hybrid cards are 5,coming from 2 + 3, and 4, from 3 + 1 .
This product is the product of the monomial basis of QSym. Con-sequently, it appears in many contexts related to it, such as the multi-zeta values, the Hurwitz multizeta functions and the multitangent functions(see [6]).
The multiset she(ααα;βββ), where ααα and βββ are sequences in Ω?, is defined tobe the set of all monomials that appear in the non-commutative polynomialααα βββ, counted with their multiplicity.
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2.4. Symmetrelity
When the alphabet Ω is an additive semi-group and A an algebra, wedefine a symmetrel mould Me• to be a mould of M•A(Ω) satisfiying for all
(ααα;βββ) ∈(Ω?)2
:MeαααMeβββ =
∑γγγ∈Ω?
〈ααα βββ|γγγ〉Meγγγ =∑
γγγ∈she(ααα,βββ)
Meγγγ .
Me∅ = 1 .
(18)
The symmetrelity imposes a strong rigidity since such properties imposea lot of relations, called the stuffle relations.
Example 2. If (x; y) ∈ Ω2 and Me• denote a symmetrel mould, then wehave necessarily:
MexMey = Mex,y +Mey,x +Mex+y. (19)
Mex,yMey = Mey,x,y + 2Mex,y,y +Mex+y,y +Mex,2y . (20)
Nevertheless, there are some stability properties about symmetrelity.
Proposition 1. Let us suppose that the target algebra A is commutative.Then, for any symmetrel moulds Me• and Ne• of M•A(Ω), the productPe• = Me• ×Ne• is also a symmetrel mould of M•A(Ω).
Let us notice that this stability property does not hold when the targetalgebra A is not commutative.
Proof. From a mould M• ∈M•A(Ω), one can consider the noncommutativeseries
m =∑ω∈Ω
Mωω ∈ A〈〈Ω〉〉 . (21)
Here, A〈〈Ω〉〉 is the algebra of all noncommutative series constructed overthe alphabet Ω, whose product is the concatenation product. It turns outthat A〈〈Ω〉〉 is also a Hopf algebra whose co-product is the dual of the stuffleproduct (see [50], chap. 1 for instance)
From this, it is easy to see that the mould M• is symmetrel if, and onlyif, the series m is a group-like element of A〈〈Ω〉〉.
The conclusion now comes from the fact that the group-like elementsof a Hopf algebra form a group. Details can be found, for instance, in [2],section 2.2.
The definition of symmetrelity may also apply to a mould defined onlyon a subset D of Ω? (in which case, we require D to be stable by the stuffleproduct).
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Lemma 3. The mould He•+ and He•− are symmetrel respectively on S? and←−S? = s ∈ N?1 ;
←−s ∈ S? .
Proof. This is a direct consequence of Lemma 1 of [6], or of [32].
As a concluding remark on these reminders on symmetrelity, we willalways write in bold, italic and underlined the vowel that indicates not onlya symmetry of the considered moulds, but also the nature of the products ofsequences that will appear. Using this, it will become simpler to distinguishwhen a mould is symmetrel or not.
2.5. Alternelity
When the alphabet Ω is an additive semi-group and A an algebra, wedefine an alternel mould Men• to be a mould of M•A(Ω) satisfying for allnonempty sequences ααα and βββ constructed over Ω?:∑
γγγ∈she(ααα ;βββ)
Menγγγ = 0 . (22)
In the same way as for symmetrelity, this sum is a shorthand: eachelement γγγ ∈ she(ααα ; βββ) have to be counted with its multiplicity.
The alternelity, as well as the symmetrelity, imposes a strong rigidity.
Example 3. If (x; y) ∈ Ω2 and Me• denote an alternel mould, then wehave necessarily:
Mex,y +Mey,x +Mex+y = 0 . (23)
Mey,x,y + 2Mex,y,y +Mex+y,y +Mex,2y = 0 . (24)
The definition of alternelity may also apply to a mould defined only ona subset D of Ω? (in which case, we require D to be stable by stuffle).
As an example of alternel mould, we have the following
Proposition 2. Let D be a mould derivation, i.e. a derivation for themould product.Then, if a symmetrel mould Me• ∈M•A(Ω) satisfies D(Me•) = Aen•×Me•,the mould Aen• is naturally alternel .
Proof. Note that, if t is a formal parameter, M•A(Ω) naturarly injectsinto M•A[[t]](Ω). Moreover, At = etD is a well-defined operator of A[[t]] for
the Krull topology. Consequently, At(Me•) is well-defined, for any mouldMe• ∈M•A(Ω), and is valued in M•A[[t]](Ω).
Since D is a mould derivation, At = etD is a mould automorphism (forthe mould product). Therefore, applying the automorphism At to both sides
10
of the equalities given by the symmetrelity of Me•, we see that the mouldAt(Me•) turns out to be a symmetrel mould.
Consequently, Ae•t = etD(Me•) × (Me•)×−1 is a symmetrel mould, ac-cording to Property 1.
Thus, for any sequences u and v of Ω?, we have:
d Aeut
dt·Aevt +Ae
ut ·
d Aevt
dt=
∑w∈she(u,v)
d Aewt
dt(25)
Since Aen• =d
dt(Ae•t )
∣∣∣∣t=0
and Ae•0 = 1•, we deduce from (25):
∑w∈she(u,v)
Aenw = Aenu Aev0 +Ae
u0 Aenv = Aenu 1v + 1u Aenv . (26)
Finally, for any nonempty sequences u and v of Ω?, we obtain∑w∈she(u,v)
Aenw = 0 , (27)
which proves the alternelity of Aen• .
2.6. Notations
To conclude this section, let us introduce two notations on sequences.Let s ∈ N?1.
• If s is a sequence of lenght r, we will use the condensed notations1..k = s1 + · · ·+ sk for all k ∈ [[ 1 ; r ]].
• ||s|| denotes the sum of all the elements of s.
3. A fundamental lemma
3.1. The operators τ , ∆+ and ∆−
We know that Hurwitz multizeta functions are a “translation” of multi-zeta values. Therefore, it is natural to examine how the shift operator actson such functions.
For p ∈ Z, let Z≤p = k ∈ Z ; k ≤ p . From now, let us consideroperators τ : H
(C − Z≤−1
)−→ H
(C − Z≤−2
), ∆+ : H
(C − Z≤−1
)−→
H(C− Z≤−1
)and ∆− : H
(C− Z≤−1
)−→ H
(C− Z≤0
)defined by :
τ(f)(z) = f(z + 1) .∆+(f)(z) = f(z + 1)− f(z) .∆−(f)(z) = f(z − 1)− f(z) .
(28)
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3.2. The action of τ on Hurwitz multizeta functions
The next lemma is a fundamental one and will allow us to express theaction of ∆+ and ∆− on a Hurwitz multizeta function.
Lemma 4. Let Js, for s ∈ N?1, be the function defined by:
Js(z) =
1
zs1, if l(s) = 1 .
0 , otherwise .(29)
Then:τ(He•+) = He•+ − τ(He•+ × J•) . (30)
Proof. For all sequences s ∈ N?1 with length r and z ∈ C − Z?≤−1 , wesuccessively have:
Hes+(z + 1) =∑
0<nr<···<n1
1
(n1 + z + 1)s1 · · · (nr + z + 1)sr
=∑
1<nr<···<n1
1
(n1 + z)s1 · · · (nr + z)sr
=
( ∑0<nr<···<n1
−∑
1=nr<···<n1
)(1
(n1 + z)s1 · · · (nr + z)sr
)= Hes+(z)− 1
(z + 1)srHes
<r
+ (z + 1) .
We can rewrite this last equality with the following compact form:
τ(He•+) = He•+ − τ(He•+
)× τ(J•). (31)
Because τ is an automorphism, this concludes the proof.
3.3. The action of ∆+ on Hurwitz multizeta functions
Because it is a discrete analogue of the classical derivative, it is tra-ditional to focus on the operator ∆+ = τ − Id. We can remark that itis a (τ ; Id)-derivative as well as a (Id ; τ)-derivative2. We then have thefollowing property explaining its action on a Hurwitz multizeta function:
2Let us remind that a (σ, τ)-derivation d defined on an algebra A is a linear map fromA to A such that, forall (a, b) ∈ A2:
d(a · b) = σ(a) · d(b) + d(a) · τ(b) .
12
Proposition 3. Let Ks, for s ∈ N?1, be the function defined by:
Ks(z) =
(−1)l(s)
(z + 1)||s||. , if l(s) 6= 0 .
0 , otherwise .
(32)
Then:∆+(He•+) = He•+ ×K• . (33)
Proof. The previous lemma gives us: τ(He•+) = He•+ − τ(He•+ × J•
).
Using it iteratively, we therefore obtain:
τ(He•+) = He•+ −He•+ × τ(J•)
+ τ(He•+
)× τ(J•)2
...
= He•+ ×(∑k≥0
(−1)kτ(J•)k)
So:
∆+(Hes+) = He•+ ×(∑k≥1
(−1)kτ(J•)k)
. (34)
We can notice that the right factor of this product is actually a locallyfinite sum, that is to say only a finite number of terms intervene in itsevaluation on a sequence s. More precisely, if s ∈ N?1 is of length r = l(s) > 0,we have:(∑
k≥1
(−1)kτ(J•)k)s
= (−1)r(τ(J•)r)s
= (−1)rτ(Js1)· · · τ
(Jsr)
=(−1)r
(z + 1)s1+···+sr = Ks(z) .
(35)Finally, if s = ∅, this sum equals 0 = Ks , which ends the proof.
Example 4. We successively have:
τ(He3,2,1+ ) = He3,2,1
+ −(τ(He•+
)× τ(J•))3,2,1
= He3,2,1+ −
(He•+ − τ
(He•+
)× τ(J•))3,2 · τ(J1
)= He3,2,1
+ −He3,2+ · τ
(J1)
+He3+ · τ
(J2)· τ(J1)
Therefore: ∆+(He3,2,1+ ) = He3,2
+ ·K1 +He3+ ·K2,1 .
Let us notice that the mould K• could also be expressed by:
K• = −τ(J•)×(
1 + τ(J•))×−1
. (36)
13
3.4. The action of ∆− on Hurwitz multizeta functions
The ∆+ derivative is complicated because the result involves severalHurwitz multizeta functions. We will prefer ∆− to it, and this operator willbecome for all this article our discrete analogue of the classical derivative.
Recall that the mould J• has been defined, for s ∈ N?1, by:
Js(z) =
1
zs1, if l(s) = 1 .
0 , otherwise .(37)
Proposition 4. We have: ∆−(He•+) = He•+ × J• .
Proof. Lemma 4 gives us: τ(He•+) = He•+ − τ(He•+ × J•
). Composing
this identity by τ−1, we therefore obtain:
He•+ = τ−1(He•+
)−He•+ × J• , (38)
This gives the result.
Example 5. Since we have τ(He3,2,1+ ) = He3,2,1
+ −(τ(He•+
)× τ(J•))3,2,1
,we successively deduce:
τ−1(He3,2,1+ ) = He3,2,1
+ +(He•+ × J•
)3,2,1= He3,2,1
+ +He3,2+ · J1
Therefore: ∆−(He3,2,1+ ) = He3,2
+ · J1 .
4. Resurgence theory and Hurwitz multizeta functions
This section is devoted to the study of the resurgent properties of Hur-witz multizeta functions. Reminders are available in Sections 4.1 and 4.2.A first example of the use of this theory is given in Section 4.3. The aim ofthis section is to prove Theorem 1 and Theorem 2.
References for the reminders are [19] in general, the book [13], the intro-duction of [4], the introductory articles [11], [16], [17], [38], [49], [51], [53] aswell as the new introductory course focused on this subject [54].
4.1. A few reminders on the Borel transform and Borel resummation process
4.1.1. Borel transform
The formal Borel transform B is defined by:
B : z−1C[[z−1]]−→ C[[ζ]]∑
n≥0
cnzn+1
7−→∑n≥0
cnn!ζn .
(39)
14
Let us first remind the terminology and the traditional notations. Aformal power series ϕ, near infinity, will systematically be denoted with atilde (˜), while the notation for its formal Borel transform will have a hat(): ϕ = B(ϕ). The variable near infinity will also systematically be denotedby z while the variable of its formal Borel transform will be ζ. We say weare working in the “formal model” when we are dealing with formal powerseries, i.e. with the z variable; when we are dealing with Borel transforms,i.e. with the ζ variable, we say we are working in the “convolutive model”(we will see why in the next proposition).
Moreover, let us denote by , · and ∂z the composition, the multiplicationand the derivative with respect to z. Finally, we define l : C −→ C to be the1-unit translation ( i.e. l(z) = z + 1) and ? to be the convolution of formalpower series defined by:
( +∞∑n=0
ann!ζn)?( +∞∑n=0
bnn!ζn)
=+∞∑n=0
1
(n+ 1)!
( n∑p=0
apbn−p
)ζn+1 , (40)
which extends the classical integral definition to formal power series.Using these notations, we can remind the classical properties of the for-
mal Borel transform (see. [38]):
Proposition 5. Let (ϕ ; ψ) ∈(z−1C
[[z−1]])2
.
Then : 1. B(∂zϕ)(ζ) = −ζϕ(ζ) .2. B(ϕ l)(ζ) = e−ζϕ(ζ) and B(ϕ l−1)(ζ) = e+ζϕ(ζ) .
3. B(ϕ · ψ)(ζ) =(ϕ ? ψ
)(ζ) .
Moreover, it is also easy to characterize the series of z−1C[[z−1]]
whoseimage under the Borel transform are analytical germs at the origin. Theirset is denoted by z−1C
[[z−1]]
1and these series called 1-Gevrey formal series.
Definition 1. A formal series near infinity ϕ(z) =∑n≥0
cnzn+1
is an element
of z−1C[[z−1]]
1, i.e. has a Borel transform which is an analytical germ at the
origin, if, and only if, there exist two positive constants (C0, C1) such that:
∀n ∈ N , |cn| ≤ C0Cn1 n! . (41)
We can conclude these reminders by the following
Proposition 6. B(z−1C
[[z−1]]
1
)= Cζ .
In other terms, B realizes an isomorphism between z−1C[[z−1]]
1⊂ z−1C
[[z−1]]
and Cζ.
15
4.1.2. Laplace transform
The Laplace transform Lθ in the direction θ ∈ R is the linear operatordefined by:
Lθ(ϕ)(z) =
∫ eiθ∞
0ϕ(ζ)e−zζ dζ . (42)
This is well-defined over the set of analytical functions ϕ on an open set ofC containing ζ ∈ C ; arg ζ = θ that have at most exponential growthin the direction θ. This last condition means that there exist two positiveconstants C and τ such that for all r > 0:∣∣∣ϕ(reiθ)
∣∣∣ ≤ Ceτr . (43)
Since1
zn+1= Lθ
(ζn
n!
)(z), for z ∈ C such that <e (zeiθ) > 0, the Borel
transform can be seen as a formal inverse of the Laplace transform, whereformally means here that we formally permute the symbols sum and inte-gral. This is actually licit for entire functions of exponential type in everydirection.
Dealing with an analytical function ϕ defined over an open set containingζ ∈ C ; arg ζ = θ which has at most an exponential growth in the directionθ (with an associated constant τ) allows us to use the theorem about theholomorphy of a parameter-dependent integral. This proves that Lθ(ϕ) is aholomorphic function in the half-plane
z ∈ C ; <e
(eiθz
)> τ
.
Figure 1: Image of the analycity domain under the Laplace transform.
4.1.3. Borel summation process
The Borel transform and Laplace transform interact to define what isnow called the Borel summation process:
16
Definition 2. For a given θ ∈ R, we say that ϕ ∈ z−1C[[z−1]]
is Borel-summable in the direction θ, and we denote ϕ ∈ SB,θ, when the followingconditions are satisfied:
1. ϕ = B(ϕ) can be analytically extended on a neighborhood Ω of eiθR+ .
2. there exist two positive constants C and τ such that for all ζ ∈ Ω ,
|ϕ(ζ)| ≤ Ceτ |ζ| . (44)
In this case, the corresponding Borel sum, denoted by Sθ(ϕ), is defined by:
Sθ(ϕ) = Lθ (B(ϕ)) . (45)
Thus, such a sum is automatically an analytic function on the half-planePθ(τ) = z ∈ C ; <e (eiθz) > τ .
This process is satisfactory, in the following sense:
1. If ϕ ∈ Cz−1
is well-defined on a neighborhood Ω of infinity, then,the Borel sums Sθ(ϕ) coincide with ϕ on Ω, for all directions θ ∈ R .
2. Sθ : SB,θ −→⋃τ>0
H(Pθ(τ)
)is an injective homomorphism which
commutes with the derivation3.
3. If ϕ ∈ SB,θ, then ϕ is the asymptotic expansion, near infinity, of Sθ(ϕ) .
To summarize, we have the diagram on Figure 2 .
4.1.4. Sectorial resummation
A formal power series ϕ is said to be uniformly Borel-summable in theinterval of direction ]θ1; θ2[ when the following conditions are satisfied:
1. ϕ can be analytically extended to a neighborhood of Ω = z ∈ C ; θ1 <arg z < θ2 .
2. there exist two positive constants C and τ such that for all ζ ∈ Ω ,
|ϕ(ζ)| ≤ Ceτ |ζ| . (46)
For such a formal power series, when we apply the Borel summationprocess, it is possible to vary the direction of the summation between θ1
and θ2. A natural question is then to understand the links between thedifferent Borel sums. The residue theorem gives the answer: they are mutualanalytical extensions.
Therefore, we can define a Borel sum, denoted by S[θ1; θ2], on the angularsector
z ∈ C ; −θ2 −π
2< arg z < −θ1 +
π
2
(47)
3H(U) denotes classicaly the space of holomorphic functions on the open subset U of C.
17
Form
al
mod
el:
ϕ(z
)=
+∞ ∑ n=
0
c nzn
+1∈z−
1C[[ z−
1]] ∩S
B,θ
.
(for
mal
pow
erse
ries
,B
orel
sum
mab
lein
the
dir
ecti
onθ)
.
B//
Convolu
tive
mod
el:
ϕ(ζ
)=
+∞ ∑ n=
0
c n n!ζn
.
(fu
nct
ion
that
can
be
anal
yti
call
yex
ten
--d
edon
an
eigh
bor
hood
ofeiθR
+an
dth
ath
asat
mos
tex
pon
enti
algr
owth
).
Geom
etr
icm
od
el:
Sθ(ϕ
)(z)
=Lθϕ
(z)
=
∫ eiθ∞
0ϕ
(ζ)e−ζzdζ
.
(an
alyti
cal
fun
ctio
nd
efin
edon
the
hal
f-p
lan
ez∈C
;<e
(eiθz)>
0)
.
Lθ
asy
mp
toti
cex
pan
sion
nea
rin
fin
ity.
TT
Fig
ure
2:
Bore
lsu
mm
ati
on
pro
cess
inth
edir
ecti
onθ
.
18
by glueing the Borel sums obtained when θ varies between θ1 and θ2 (seeFigure 3), because two Borel sums, corresponding to angles satisfying (47)coincide on the intersection of their definition domains (thanks to contourintegrals of holomorphic functions).
This is true as long as ϕ has no singularities in the sector Ω . Otherwise,the residue theorem shows that the difference between Lθ−(ϕ) and Lθ+
(ϕ),the Borel sums just before and just after a singularity in the direction θ, donot coincide any more: this is the so-called Stokes phenomenon.
Figure 3: Sectorial resummation.
4.2. Reminders on resurgence theory
As we have just seen, we need to have a precise knowledge of the dif-ferent singularities of ϕ = B(ϕ) to compare the Borel sectorial sums of thedivergent series ϕ, which is supposed to be uniformly Borel-summable (inan interval) .
Since we will only deal with simple resurgent functions in the sequel, weonly remind the definition of this class of functions. For more information,we refer the reader to Ecalle’s text (the three volumes of [19]) , or to theintroductory article/book [51] and [13].
4.2.1. Simple singularities
We say that a function ϕ, defined and holomorphic on an open set D, hasa simple singularity at ω ∈ C adherent to D when ϕ can be expanded nearω as the sum of a logarithmic singularity, a polar singularity and a regularpart. Thus, there exist C ∈ C and two germs of holomorphic functionsΦ and reg such that, in a neighborhood U of ω, the following equality issatisfied:
ϕ(ζ) =(ζ−→ ω)
C
2iπ(ζ − ω)+
1
2iπΦ(ζ − ω) log(ζ − ω) + reg(ζ − ω) . (48)
19
The number C is called the residue of ϕ and Φ is called the variation ofϕ. These objects can be computed from ϕ, independently from the choiceof the branch of the logarithm, by:
C = 2iπ limζ−→ωζ∈U
(ζ − ω)ϕ(ζ) , Φ(ζ) = φ(ω + ζ)− φ(ω + ζe−2iπ
), (49)
where φ(ω + ζe−2iπ
)means the evaluation at the point ω+ζ of the analytic
continuation of φ following the path t 7−→ ω + ζe−2iπt, t ∈ [0; 1].(48) is usually denoted in a simpler form:
singωϕ = Cδ + Φ ∈ Cδ ⊕ Cζ . (50)
Here, the operator singωϕ denotes the asymptotic expansion near thepoint ω of ϕ, while δ denotes a Dirac. Let us remind that the algebra(Cζ, ?) has no unit. We add a formal one by an extension of the Boreltransform:
B : C[[z−1]]1∼−→ Cδ ⊕ Cζ , (51)
whereδ = B(1) . (52)
4.2.2. Endlessly continuable germs
We say that a holomorphic germ ϕ at the origin is an endlessly con-tinuable germ when, for any finite broken line L, there exists a finite setΩL ⊂ L of singularities such that ϕ has an analytic continuation along allthe possible paths obtained by following L and getting around each point ofΩL turning left or right.
4.2.3. S-resurgent functions
We define here the notion of simple resurgent function, which is a par-ticular case of a more general notion (see [11], [13], [16], [19], [51]).
In the convolutive model, S-resurgent functions are defined to be end-lessly continuable holomorphic germs at the origin, with simple singularities.
In the formal model, S-resurgent functions are formal power series inz−1C
[[z−1]]
such that their Borel transform is an S-resurgent function in theconvolutive model. Let us remark that, necessarily, an S-resurgent functionin the formal model is a 1-Gevrey series.
We will denote by RESsimple
and RESsimple
the set of simple resurgentfunctions in the formal and convolutive models.
The important point is the stability of RESsimple
(resp. RESsimple
) underthe ordinary product (resp. convolution product) of formal power series (see[52]).
20
4.2.4. Alien derivations
From now on, we will restrict to a particular case of S-resurgent func-tions, those dealing with Borel transform singularities located in Ω = 2iπZ∗ .
The alien derivatives are linear operators acting on the resurgent func-tions that “measure” the singularities near each ω ∈ Ω. Let us emphasizethat they indeed define derivations (for the usual product in the formalmodel or for the convolution product in the convolutive model), since theycome from the logarithm of the Stokes automorphism.
To make it explicit in the convolutive model, let us considerω = 2imπ ∈ Ω . Given εεε = (ε±1; · · · ; ε±(m−1)) ∈ +1;−1|m|−1,we define the path γ(εεε) constructed by following ]0;ω[ and gettingaround the intermediate singularities 2ikπ, ±k ∈ [[ 1 ; |m| − 1 ]] onhalf-circles, the k-th being oriented on the left side of the imagi-nary axis when εk = −1 and oriented on the right side of the sameaxis when εk = +1 .
Thus, the alien derivative ∆∆ω acts on ϕ ∈ RESsimple
by
∆∆ω(ϕ) =∑
εεε=(ε±1;··· ;ε±(m−1))∈+1;−1|m|−1
p(εεε)! q(εεε)!
m!singω(contγ(εεε)ϕ) .
In this formula, p(εεε) and q(εεε) stand respectively for the number of signs+ and − in εεε . Moreover, contγ(εεε) is the analytic continuation of ϕ alongthe path γ(εεε).
In the formal model, we define the linear operator
∆∆ω : RESsimple
7−→ RESsimple
(53)
by the following commutative diagram:
RESsimple ∆∆ω //
B
RESsimple
B
RESsimple ∆∆ω // RES
simple
Proposition 7. If f and g are two simple resurgent functions, then h(z) =f(z + g(z)) is also a resurgent function, and we have:
∆∆ωh(z) = e−ωg(z)(
∆∆ωf)
(z + g(z)) + (∂f)(z + g(z)) ·∆∆ω g(z) . (54)
Proof. See, for instance [19], Vol. 1, Section 2e, or [54], Section 30.
21
4.3. On a generic 1-order difference equation
Examples of the use of Borel-Laplace resummation can be found in theliterature. For example, we refer the reader to the third volume of [19](p. 243 to 246) to see a resurgent approach of the function log Γ, to [16] forthe Euler equation ϕ′(z) + ϕ(z) = z−1 or to [17] and [49] for a resurgentstudy of the Airy function.
We recall here the use of the Laplace transform for a simple 1-orderdifference equation (see [4] and [51]). This equation will be the prototype ofthe equations we will deal with in the next section.
Let us fix a holomorphic germ a ∈ z−2Cz−1
and study the 1-orderdifference equation:
ϕ(z − 1)− ϕ(z) = a(z) . (55)
We will study it first when the unknown ϕ is a formal power series at infinity,then construct solutions which are holomorphic functions (expressed as aLaplace transform) which turn out to have explicit expressions dependingon the germ a .
4.3.1. Preliminary upper bounds
Since a(z) =∑n≥1
anzn+1
∈ z−2Cz−1, there exist two positive constants
C0 and C1 such that
|an| ≤ C0C1n , for all n ∈ N∗ . (56)
Consequently, we have:∣∣a(ζ)∣∣ ≤ C0
(eC1|ζ| − 1
)≤ C0|ζ|eC1|ζ| for all ζ ∈ C . (57)
Moreover, we have the following
Lemma 5. If ζ = reiθ, with θ ∈]−π
2;π
2
[∪]π
2;3π
2
[, we have:
∣∣∣∣ ζ
eζ − 1
∣∣∣∣ ≤ e|ζ|
| cos θ|. (58)
Proof. If θ ∈]−π
2;π
2
[,
∣∣∣∣ ζ
eζ − 1
∣∣∣∣ ≤ |ζ| · |e−ζ | ·∣∣∣∣ 1
1− e−ζ
∣∣∣∣≤ re−r cos θ 1
1− e−r cos θ
≤ 1
cos θ
r cos θ
er cos θ − 1≤ 1
cos θ
≤ e|ζ|
cos θ.
22
If θ ∈]π
2;3π
2
[,
∣∣∣∣ ζ
eζ − 1
∣∣∣∣ ≤ |ζ| ·∣∣∣∣ 1
1− eζ
∣∣∣∣ ≤ r
1− er cos θ
≤ 1
| cos θ|e−r cos θ −r cos θ
e−r cos θ − 1≤ er
| cos θ|
≤ e|ζ|
| cos θ|.
4.3.2. The resurgent character of the formal solution
Using the Borel transform, if ϕ is a solution of (55), we naturally obtain:
1. eζϕ(ζ)− ϕ(ζ) = a(ζ) , which implies ϕ(ζ) =a(ζ)
eζ − 1.
2. ϕ defines a meromorphic function on C of exponential type in all di-rections θ ∈
]−π
2 ; π2[∪]π2 ; 3π
2
[, with poles that can only be located in
2iπZ∗, since a is an entire function vanishing at the origin satisfying:
|ϕ(ζ)| =∣∣∣∣ a(ζ)
eζ − 1
∣∣∣∣ ≤ C0
| cos θ|e(C1+1)|ζ| , for all ζ ∈ C− iR , (59)
according to (57) and Lemma 5.
Thus, Equation (55) has a unique solution in z−1C[[z−1]]
ϕ = B−1
(a(ζ)
eζ − 1
), (60)
which is an S-resurgent function and satisfies:
∆∆ωϕ = a(ω) δ (61)
for all ω ∈ 2iπZ∗ . This explains the term of resurgent functions, since wecan conceptually write (even if ϕ(ω) is infinite) ∆∆ωϕ = (eω − 1)ϕ(ω) δ, andthus ϕ “reappears” in the singularity ω .
4.3.3. Sectorial resummations
Moreover, ϕ is Borel-summable in all directions θ ∈]−π
2;π
2
[∪]π
2;3π
2
[.
The sectorial resummation principle gives us two analytical functions, ϕe andϕw, which, by construction, are two resurgent functions in the geometricmodel and defined respectively on the east half-plane and west half-planeby:
ϕe = Lθ(a(ζ)
eζ − 1
)= Lθ(ϕ), θ ∈
]−π
2;π
2
[
ϕw = Lθ(a(ζ)
eζ − 1
)= Lθ(ϕ), θ ∈
]π
2;3π
2
[.
(62)
23
Denoting by d(z;F ) the distance of z ∈ C to the set F and taking intoaccount the properties of the Laplace transform, there exists τ > 0 suchthat these two solutions are analytical on the domains De(τ) and Dw(τ)(see Figure 4) defined by De(τ) = C−z ∈ C ; d(z;R−) ≤ τ and Dw(τ) =C− z ∈ C ; d(z;R+) ≤ τ , according to Equation (59).
Holomorphic domain of Holomorphic domain ofthe sectorial resummation ϕw . the sectorial resummation ϕe .
Figure 4: Illustration of the complementary set of De(τ) and Dw(τ).
4.3.4. Characterization of ϕe and ϕwIt is not difficult to see that ϕe|R>τ is a solution of (55) which is defined,
by construction, on R>τ = x ∈ R ; x > τ and admits 0 as limit near +∞(since ϕe is a Laplace transform). Actually, such a solution turns out to beunique: if ψ is another solution of (55) defined on R>τ ′ with 0 as limit near+∞, ψ−ϕe becomes a 1-periodic function defined on R>max(τ,τ ′) which alsohas 0 as limit near +∞; this necessarily imposes the equality ψ = ϕe inR>max(τ,τ ′).
Likewise, ϕw |R−<−τ, where R−<−τ = x ∈ R ; x < −τ, is the unique
solution of (55) defined on a set R−<−τ and admiting 0 as limit near −∞.
On the other side, from Equation (56), we deduce that
|a(z)| ≤ C0C1
|z|(|z| − C1), if z ∈ C−De(C1) . (63)
Moreover, it is clear, if z ∈ C−De(C1), where C1 is defined by Equation (56),that we have z + k ∈ C−De(C1) for all k ∈ N∗. In particular, |z + k| > C1.Therefore,
|a(z + k)| ≤ C0C1
|z + k|(|z + k| − C1), if z ∈ C−De(C1) and k ∈ N∗ . (64)
24
Moreover, if K is a compact subset of C−De(C1), and if a > 0 is such thatK ⊂ D(0, a), we obtain:
|a(z + k)| ≤ C0C1(√k(k − 2a) + C2
1 − C1
)2 , (65)
if z ∈ K ⊂⊂ C−De(C1) and k ∈ N∗>2a, according to |z+k| ≥√k2 − 2ak + C2
1
for all z ∈ C−De(C1) and k ∈ N∗.Consequently, z 7→
∑k>0
a(z + k) is a normally convergent series on every
compact subset of C − De(C1) of holomorphic functions on C − De(C1).Thus, it defines a holomorphic function over C−De(C1).
Using simillar arguments with C−Dw(C1), we see that z 7→ −∑k≤0
a(z + k)
is also a holomorphic function of C−Dw(C1).
Moreover, it is clear that z 7→∑k>0
a(z + k) and z 7→ −∑k≤0
a(z + k) are
two solutions of (55). The limit of these at +∞ and −∞ respectively is 0.Consequently, they coincide with ϕe and ϕw on a set which type is R>τ andR<−τ for a certain constant τ > 0. According to the analytic continuationprinciple, we therefore have proven the following
Lemma 6. Let a ∈ z−2Cz−1.1. The difference equation ϕ(z − 1)− ϕ(z) = a(z) admits a unique solutiondefined on a set R>τ , for a certain constant τ > 0, with 0 as limit near +∞,given by:
ϕe(z) = L0
(a(ζ)
eζ − 1
)=∑k>0
a(z + k) . (66)
It turns out that ϕe is holomorphic on C−De(τ).2. The difference equation ϕ(z − 1)− ϕ(z) = a(z) admits a unique solutiondefined on a set R<−τ , for a certain constant τ > 0, with 0 as limit near−∞, given by:
ϕw(z) = Lπ(a(ζ)
eζ − 1
)= −
∑k≤0
a(z + k) . (67)
It turns out that ϕw is holomorphic on C−Dw(τ).
25
4.3.5. Stokes phenomenon
Since De∩Dw has two connected components, which are z ∈ C ; =mz >τ and z ∈ C ; =mz < −τ, we can evaluate the difference ϕe − ϕw ineach of these connected components. Then, the residue theorem gives us forz such that =mz < −τ :∫
γϕ(ζ)e−zζ dζ =
m∑k=1
2iπRes
(z 7−→ a(ζ)e−ωz
eζ − 1; 2ikπ
)
=
m∑k=1
2iπa(2ikπ)e−2ikπz ,
(68)
where γ is the path described on Figure 5.
Figure 5: An integration path.
Besides, it is easy to see that the limit of the previous integral on thepath restricted to the arc of circle is 0 as R approaches +∞. The series∑k>0
2iπa(2ikπ)e−2ikπz is a convergent one when =mz < −τ . Thus, when
R approaches +∞, we obtain:
∀z ∈ C , =mz < −τ , ϕe(z)− ϕw(z) =∑
ω∈2iπN∗2iπa(ω)e−ωz , (69)
which expresses the Stokes phenomenon in the south part of the complexplane.
In the same way, but with a symmetric contour relative to the real axis,we obtain:
∀z ∈ C , =mz > τ , ϕe(z)− ϕw(z) =∑
ω∈−2iπN∗2iπa(ω)e−ωz , (70)
which expresses the Stokes phenomenon in the north part of the complexplane.
4.3.6. Conclusion of the solution of the generic 1-order difference equation.
The difference equation
ϕ(z − 1)− ϕ(z) = a(z) ,
where a ∈ z−2Cz−1
is a holomorphic germ near infinity, has:
26
• a unique formal solution ϕ in z−1C[[z−1]]
which turns out to be S-resurgent.
• a unique solution ϕ in the convolutive model which is also S-resurgent.
• two analytical solutions ϕe and ϕw, defined respectively as the sec-torial resummations of ϕ in the east and west directions which havean asymptotic expansion near infinity (in those respective directions)which is nothing but ϕ.
4.4. On the resurgence of Hurwitz multizeta functions
In this section, we will use recursively the resurgent study of the generic1-order difference equation made in the previous section, in order to gener-alize it to a resurgent study of the 1-order mould difference equation
∆−(M•) = M• × J• . (71)
Let us notice that any solution of this mould equation, with perhap an-other condition (such as a given limit near infinity or the absence of constantterm), will turn out to be a symmetrel mould. Consequently, to remind usof this property, we are going to add systematically the letter e to the nameof the different types of solutions we will consider.
This resurgent study of (71) will lead us to define four moulds, relatedto Hurwitz multizeta functions, in the same way that we have found foursolutions of the generic 1-order difference equation:
He•
will denote a mould of formal solutions of (71).
He•
will denote the Borel transform of any formal solution He•
of(71):
Hes
= B(He
s). (72)
He•e and He•w, defined respectively, if possible, as the sectorial resum-
mations of He•
in the east and west directions.
In order to always have ∆−(M s) ∈ z−2C[[z−1]], we will define thesemoulds on the subset S? of N?1 of sequences s of positive integers such thats1 ≥ 2.
In order to be consistent with the previously introduced notations, letus mention that all these four notations might have a subscript ‘+’ that wehave omitted throughout all the rest of this section for simplicity. Neverthe-less, we still use the subscript ‘+’ or ‘-’ to specify which Hurwitz multizetafunction we are currently dealing with.
These notations and restrictions being introduced, we now are going toprove the following
27
Theorem 1. 1. There exists a unique mould He•, defined over the subset
S? of N?1, of formal series near infinity which are solutions of the moulddifference equation ∆−(M•) = M•× J• . These functions turn out tobe:
• formal S-resurgent functions.
• Borel-summable formal series in all directions θ ∈]−π
2;π
2
[and
θ ∈]π
2;3π
2
[.
2. There exist two unique moulds He•e and He•w of analytical functionsdefined respectively on the east half complex plane and the west halfcomplex plane ( i.e. on domains such as those shown on Figure 4),that admit 0 as a limit near infinity, but in the real axis: they are themoulds of the sectorial sums of He
•in the geometric model and turn
out to be moulds valued in the algebra of resurgent functions.
3. The Stokes phenomenon obtained from the sectorial sums can be ex-pressed by:
He•e × (He•w)×−1 = T e• , (73)
where T e• denotes the mould of multitangent functions.
4.4.1. Reminders on the multitangent functions
The mould T e• denotes the mould of multitangent functions and is de-fined for all sequences s of positive integers satisfying s1 ≥ 2 and sr ≥ 2 (ifs is of length r) and for all z ∈ C− Z by:
T es1,··· ,sr(z) =∑
−∞<nr<···<n1<+∞
1
(n1 + z)s1 · · · (nr + z)sr. (74)
These multitangent functions have been extensively studied in [6] froma combinatorial, algebraic and analytic point of view. It results from thisstudy that the multitangent functions are a nice functional generalization ofthe multizeta values.
The multitangents satisfy not only the stuffle relations (defined in Sec-tion 2.4) but also some quite mysterious nontrivial linear relations (see [6])which are analogue in a certain sense to the other relations satisfied by themultizeta values (the shuffle relations as well as the double shuffle relations,to be precise; see [56] for an introduction).
For instance, one has:
4T e3,1,3 − 2T e3,1,1,2 + T e2,1,2,2 = 0 , (75)
which is equivalent toζ(2, 1, 1) = 4ζ(3, 1) (76)
28
andζ(2)ζ(2, 1) = ζ(2, 1, 2) + 6ζ(3, 1, 1) + 3ζ(2, 2, 1) . (77)
Note that (77) is exactly a shuffle relation for the multizeta values and (76)is a consequence of the three families of relations (the shuffle relations, thestuffle relations and the double shuffle relations) which are conjectured tospan all the algebraic relations between multizeta values (see [20] and [56]).
There is no doubt that, in comparison to the algebra of multitangentfunctions or multizeta values, the Hurwitz multizeta functions span a simplealgebra since they satisfy only algebraic relations coming from the stufflerelations, which is nevertheless an interesting one because it is isomorphic toa famous algebra in combinatorics, the algebra of quasisymmetric functionsQSym, as will be seen in Section 5.
Moreover, the mould of multitangent functions has a nice mould factor-ization in terms of Hurwitz multizeta functions He•+ and He•− which willenlighten the Stokes phenomena of Hurwitz multizeta functions.
Lemma 7. We have : T e• = He•+ × (1• + J•)×He•− .
Proof. See [6], p. 55.
4.4.2. On the case of length 0 and 1
Because we want the moulds He•, He
•, He•e and He•w to be symmetrel
moulds, we necessarily have to define their values on the empty sequence tobe the unit of the product used for their symmetrelity properties, i.e. the
ordinary product for He•, He•e and He•w, but the convolution product for
He•:
He∅
= He∅e = He∅w = 1 , He∅
= δ . (78)
The case of length 1 has already been done in Section 4.3. Let us remindwhat we have obtained. The difference equation
∆−Hes(z) =
1
zs, s ≥ 2 , (79)
has a unique formal solution in z−2C[[z−1]] which is S-resurgent:
Hes
= B−1(Hes) , where He
s(ζ) =
1
(s− 1)!
ζs−1
eζ − 1∈ ζCζ . (80)
It satisfies for all n ∈ Z∗:
∆∆2inπHes
= Res(He
s, 2inπ
)δ =
(2inπ)s−1
(s− 1)!δ = Js(2inπ)δ . (81)
29
Moreover, Hes
is Borel-summable in all the directions θ ∈]−π
2;π
2
[or
θ ∈]π
2;3π
2
[and, according to (66) and (67):
• Equation (79) has a unique solution, denoted by Hese, defined over R+
such that Hese(z) −→z−→+∞z∈R+
0:
Hese = L0(He
s)
= Hes+ . (82)
• Equation (79) has a unique solution, denoted by Hesw, defined over R−such that Hesw(z) −→
z−→−∞z∈R−
0:
Hesw = Lπ(He
s)
= −(Js +Hes−) . (83)
Finally, the Stokes phenomena are given in two different ways. The firstone uses the expression of Hese and Hesw in terms of Hes+ and Hes− and thetrifactorization of multitangent functions. The second one uses the residuetheorem, as seen in Section 4.3.
Hese(z)−Hesw(z) = Hes+(z) + Js(z) +Hes−(z) = T es(z) . (84)
Hese(z)−Hesw(z) = sg(=mz)(2iπ)s
(s− 1)!
∑k>0
ks−1e−2ikπz . (85)
Let us remind that all these quantities are well-defined since s ≥ 2.
4.4.3. The case of length 2
Let us now consider the difference equation
∆−(Hes1,s2
) = Hes1 · Js2 , (86)
with s1 ≥ 2 and s2 ≥ 1.Its formal solution is simple, according to
Lemma 8. There exists a unique formal series near infinity Hes1,s2
, whichis a solution of the difference equation (86):
Hes1,s2
= B−1(He
s1,s2), (87)
where
Hes1,s2
(ζ) =1
eζ − 1
(He
s1? Js2
)(ζ) . (88)
30
Proof. Any formal solution Hes1,s2
of (86) has a Borel transform Hes1,s2
which satisfies:
(eζ − 1)Hes1,s2
(ζ) =(He
s1? Js2
)(ζ) . (89)
Consequently, Hes1,s2
is necessarily defined by (87) and (88), which provesunicity.
Conversely, defining Hes1,s2
by (87) and (88), and using successively thesecond point of Property 5 rewritten as
B−1 (ϕ) l−1 = B−1(exp · ϕ
)(90)
and then its third point, we successively have:
∆−
(He
s1,s2)
(z) = B−1(He
s1,s2) l−1(z)− B−1
(He
s1,s2)
(z)
= B−1(exp · He
s1,s2)
(z)− B−1(He
s1,s2)
(z)
= B−1
(eζ
eζ − 1·(He
s1? Js2
))(z)
−B−1
(1
eζ − 1·(He
s1? Js2
))(z)
= B−1(He
s1? J)
(z)
= Hes1 · Js2 ,
which proves the existence of a formal solution near infinity of (86) andconcludes the proof of the lemma.
Note that:
Lemma 9. Hes1,s2
is a S-resurgent function, as well as Hes1,s2
.
Proof. As recalled in Section 4.2, the important point concerns the stabil-
ity of RESsimple
(resp. RESsimple
) under the ordinary product (resp. convo-lution product) of formal power series (see [52]).
Since Hes1
, ζ 7−→ Js2(ζ) and ζ 7−→ 1
eζ − 1are S-resurgent functions,
Hes1,s2
is a S-resurgent function in the convolutive model. Consequently,He
s1,s2is a S-resurgent function in the formal model.
We are now able to look at the Borel-summability of Hes1,s2
.
Lemma 10. ζCζ?Cζ ⊂ ζ2Cζ, where ? denotes the convolution prod-uct.
31
Proof. It is enough to prove, for all non negative integers p and q, thatζp+1 ? ζq ∈ ζ2Cζ, which is straightforward.
As a consequence of (80), ζ 7−→ Hes1? Js2(ζ) is a meromorphic func-
tion over C vanishing at the origin with poles located in 2iπZ. Therefore,He
s1,s2defines a meromorphic function on C with poles which can only be
localised in 2iπZ∗. Let us mention here that this fact will be reobtainedin Section 4.4.6 as a consequence of the computation of the alien deriva-tives done in Theorem 2. This proves its Borel summability in all directions
θ ∈]−π
2;π
2
[∪]π
2;3π
2
[if it is of exponential type in all directions. But,
this is a consequence of the exponential type of ζ −→(eζ − 1
)−1and the
fact that if f and g are of exponential type 0, it is also the case for f ? g.
So, we have proved
Lemma 11. Hes1,s2
is a Borel summable formal series in all directions
θ ∈]−π
2;π
2
[and θ ∈
]π
2;3π
2
[and gives rise to two analytical functions
Hes1,s2e and Hes1,s2w which are solutions of (86) and defined by:
Hes1,s2e (z) = L0(He
s1,s2)
and Hes1,s2w (z) = Lπ(He
s1,s2), (91)
on domains as shown in Figure 4.
Using the characterization of Lemma 6, we see that:
Lemma 12. 1. Hes1,s2e is the unique function defined over a neighbour-hood of R+ satisfying:
∆−Hes1,s2e = Hes1e · Js2
Hes1,s2e (z) −→z−→+∞z∈R+
0 (92)
and is given by
Hes1,s2e (z) =∑n2>0
Hes1e (n2 + z)Js2(n2 + z)
=∑
n1>n2>0
1
(n1 + z)s1(n2 + z)s2= Hes1,s2+ (z) (93)
2. Hes1,s2w is the unique function defined over a neighbourhood of R− sat-isfying:
∆−Hes1,s2w = Hes1w · Js2
Hes1,s2w (z) −→z−→−∞z∈R−
0 (94)
32
and is given by
Hes1,s2w (z) = −∑n2≤0
Hes1w (n2 + z)Js2(n2 + z)
= +∑
n1≤n2≤0
1
(n1 + z)s1(n2 + z)s2(95)
4.4.4. On the general case
Because the general case is proven exactly in the same way as in length 2,we only give here the recursion formulas which are useful and conclude theproof of parts 1 and 2 of Theorem 1.
Let us suppose that, for a positive integer r, we have constructed Hes1,··· ,sr
,He
s1,··· ,sr, Hes1,··· ,sre and Hes1,··· ,srw for all (s1, · · · , sr) ∈ (N∗)r such that
s1 ≥ 2.Then, the resurgent study of the difference equation
∆−(Hes1,··· ,sr+1
) = Hes1,··· ,sr · Jsr+1 , (96)
with s1 ≥ 2 and (s2, · · · , sr+1) ∈ (N∗)r, is given by:
Hes1,··· ,sr+1
(ζ) =1
eζ − 1
(He
s1,··· ,sr? Jsr+1
)(ζ) . (97)
Hes1,··· ,sr+1
(z) = B−1(He
s1,··· ,sr+1)
(z) . (98)
Hes1,··· ,sr+1e (z) = L0
(He
s1,··· ,sr+1)
(z) (99)
Hes1,··· ,sr+1w (z) = Lπ
(He
s1,··· ,sr+1)
(z) . (100)
These equations define four moulds over S?: He•, He
•, He•w and He•e.
As we have explained at the begining of Section 4.4.2, we wanted thesemoulds to be symmetrel .
Proposition 8. The mould He•
is a symmetrel mould, for the usual prod-uct of formal power series.
Proof. We have to prove that, for all sequences (u,v) ∈ (S?)2
Heu· He
v=
∑w∈she(u,v)
Hew. (101)
Let us show this proposition by an induction process on l(u)+l(v). Equation(101) is true for u = v = ∅, which allows to begin the induction process,but also when u = ∅ or v = ∅.
Therefore, let us suppose that Equation (101) is true for all sequences(u,v) ∈ (S?)2 such that l(u)+ l(v) ≤ N , with N ∈ N∗. From now on, we fix
33
two particular sequences (u,v) ∈ (S?)2 such that l(u) + l(v) = N − 1 andtwo particular letters a and b (i.e. a, b ∈ N∗). We will show that Equation(101) is valid for (u · a,v · b).
1. Using the fact that ∆− is a (Id, τ−1)-derivation and the recursivedefinition (17) of the stuffle product, we successively have:
∆−(Heu·a· He
v·b) = ∆−(He
u·a) · He
v·b+ τ−1(He
u·a) ·∆−(He
v·b)
= Heu· Ja · He
v·b+ He
u· He
v· Ja · Jb + He
u·a· He
v· Jb
=∑
w∈she(u,v·b)
Hew· Ja +
∑w∈she(u,v)
Hew· Ja+b
+∑
w∈she(u·a,v)
Hew· Jb
=∑
w∈she(u,v·b)
∆−
(He
w·a)+
∑w∈she(u,v)
∆−
(He
w·(a+b))
+∑
w∈she(u·a,v)
∆−
(He
w·b)
=
∑w∈she(u·a,v)·b
+∑
w∈she(u,v)·(a+b)
+∑
w∈she(u,v·b)·a
(∆−
(He
w))
=∑
w∈she(u·a,v·b)
∆−
(He
w)
Therefore, for all sequences (u,v) ∈ (S?)2 such that l(u) + l(v) = N − 1and two letters a and b, there exists a correction corru·a,v·b ∈ z−1C[[z−1]],which is 1-periodic and satisfies:
Heu·a· He
v·b=
∑w∈she(u·a,v·b)
Hew
+ corru·a,v·b . (102)
2. We now extend the definition of the correction corra,b to sequences(a,b) ∈ (S?)2 such that l(a) + l(b) ≤ N by corra,b = 0, so that
Hea· He
b=
∑c∈she(a,b)
Hec
+ corra,b , (103)
34
for all sequences (a,b) ∈ (S?)2 such that l(a)+ l(b) ≤ N +1. Consequently,for any sequences (a,b, c) ∈ (S?)3 satisfying l(a) + l(b) + l(c) ≤ N + 1, wehave:
Hea· He
b· He
c= He
a·
∑d∈she(b,c)
Hed
+ corrb,c
=
∑d∈she(b,c)
∑e∈she(a,d)
Hed
+ Hea· corrb,c
=∑
e∈she(a,b,c)
Hee
+ Hea· corrb,c
Hea· He
b· He
c=
∑d∈she(a,b)
Hed
+ corra,b
· Hec=
∑d∈she(a,b)
∑e∈she(c,d)
Hee
+ corra,b · Hec
=∑
e∈she(a,b,c)
Hee
+ corra,b · Hec
Consequently, l(a)+l(b)+l(c) ≤ N+1 =⇒ Hea·corrb,c = corra,b ·He
c.
Better, if l(a)+l(b)+l(c) ≤ N+1 and a 6= ∅, we necessarily have l(b)+l(c) ≤N , so corrb,c = 0, which implies for all sequences a, b, c ∈ S? that:
l(a) + l(b) + l(c) ≤ N + 1a 6= ∅ =⇒ corra,b · He
c= 0 =⇒ corra,b = 0 .
Moreover, it is clear that corra,∅ = corr∅,b = 0 if l(a) = l(b) = N + 1. Thisallow us to sum up the values of the correction corra,b:
l(a) + l(b) ≤ N + 1 =⇒ corra,b = 0 . (104)
Finally, we can update Equation (102):
Heu·a· He
v·b=
∑w∈she(u·a,v·b)
Hew, (105)
which concludes the proof of the proposition, by an induction process.
We therefore obtain the following
35
Corollary 1. 1. The mould He•
is a symmetrel mould, for the convolutiveproduct of formal power series.2. The moulds He•w and He•e are symmetrel moulds, for the usual productof holomorphic functions.
Proof. The first point is straighforward, applying the Borel transform tothe symmetrelity relations of He
•.
The second point is a direct application of the first point and the classicalformula
Lθ(f) · Lθ(g) = Lθ(f ? g) . (106)
4.4.5. The Stokes phenomenon
Now, we have to focus on the Stokes phenomenon. Note that we have noteluded its study in length 2 because it cannot be expressed as a differenceof the two Borel resummations on the east and west side. Nevertheless, itwill be reinforced.
From the characterization given in Section 4.3.4 of ϕe and ϕw, it followsthat:
Hes1,···sre (z) =∑
0<nr<···<n1<+∞
1
(n1 + z)s1 · · · (nr + z)sr. (107)
Hes1,···srw (z) =∑
−∞<n1≤···≤nr≤0
(−1)r
(n1 + z)s1 · · · (nr + z)sr. (108)
According to Lemma 2, we consequently obtain:
Lemma 13. He•w =(
(1• + J•)×He•−)×−1
.
Proof. In the sum defining He•w, we can distinguish whether the index n1
is equal to 0 or not. This gives successively:
Hes1,···srw (z) =∑
−∞<n1≤···≤nr<0
(−1)r
(n1 + z)s1 · · · (nr + z)sr
+∑
−∞<n1≤···≤nr−1≤nr=0
(−1)r
(n1 + z)s1 · · · (nr + z)sr
=(Hes1,··· ,sr−
)×−1(z)− 1
zsr
∑−∞<n1≤···≤nr−1<0
(−1)r−1
(n1 + z)s1 · · · (nr−1 + z)sr−1
+1
zsr−1+sr
∑−∞<n1≤···≤nr−2≤nr−1=nr=0
(−1)r−2
(n1 + z)s1 · · · (nr + z)sr
=...
36
=
=(Hes1,··· ,sr−
)×−1(z)− 1
zsr
(Hes1,··· ,sr−1−
)×−1(z)+
1
zsr−1+sr
(Hes1,··· ,sr−2−
)×−1(z) + · · ·+ (−1)r
zs1+···+sr .
Consequently, we have:
He•w =(He•−
)×−1 ×∑n≥0
(−J•)×n
=(He•−
)×−1 × (1• + J•)×−1
=(
(1• + J•)×He•−)×−1
.
(109)
Therefore, the Stokes phenomena are expressed multiplicatively by:
He•e × (He•w)×−1 = He•+ × (1• + J•)×He•− = T e• , (110)
which concludes the proof of Theorem 1.
Note that Theorem 1 also has an analogue for the difference equationusing ∆+ satisfied by the Hurwitz multizeta functions. The east Borel re-summation would have been the inverse of the mould He•+× (1•+J•), whilethe west Borel resummation would have been the mould He•−. Thus, theStokes phenomenon would have been expressed as the inverse of the mouldof multitangent functions.
4.4.6. On the alien derivations of He•
Let us come back in this section to the computation of the alien deriva-tives of He
•and He
•. In an explicit way, this computation is actually
difficult. But, from the mould difference equation ∆−He•
= He•× J•, it is
possible to predict some formulas.
We will see that He•
turns out to be a resurgent monomial. Such afunction, according to [21], Section 3, is a resurgent function which behaves
nicely under product (which is the case for He•
because of the symmetrelity),under the ordinary derivation (which is also the case according to (5)) as wellas under the alien derivatives. Thus, “the resurgent monomials are to thealien derivatives what the ordinary monomials are to the natural derivative”(see [21], p. 104).
37
Theorem 2. For each ω ∈ C, there exists a scalar-valued alternel mouldHen•ω defined over S∗ such that:
∆∆ωHe•
= Hen•ω × He•. (111)
Moreover, Hen•ω = 0• if ω 6∈ 2iπZ∗ .
Let us remind that such a theorem directly goes back to the origin of thename of the resurgent function and generalizes the remark made relativelyto Equation (61). Actually, computing an alien derivative of a resurgentfunction gives rise to a singular phenomenon: the resurgent function resurgesin the variation of the singularities of its analytic continuation.
Proof. Let us fix ω ∈ C .We can always define a mould Hen•ω by ∆∆ωHe
•= Hen•ω × He
•where
Hen•ω is a priori valued in C[[z−1]].According to Property 7, we have:
∆∆ω (ϕ(z − 1)) = eω (∆∆ωϕ) (z − 1) , for all ϕ ∈ RESsimple
. (112)
We can now compute in two different ways ∆∆ω
(∆−He
•):
∆∆ω
(∆−He
•)(z) = eω ∆∆ω
(He•)
(z − 1)−∆∆ω
(He•)
(z)
= (eω − 1) Hen•ω(z − 1)× He•(z − 1) + ∆−
(Hen•ω × He
•)(z)
= (eω − 1) Hen•ω(z − 1)× He•(z − 1)
+∆−
(Hen•ω
)(z)× He
•(z − 1) + Hen•ω(z)×∆−
(He•)
(z) .
∆∆ω
(∆−He
•)(z) = ∆∆ω
(He•)
(z)× J•(z) + He•(z)×∆∆ω (J•) (z)
= Hen•ω(z)× He•(z)× J•(z) + He
•(z)× B−1
(∆∆ωJ
•)(z)= Hen•ω(z)×∆−
(He•)
(z) + He•(z)× B−1(0)(z)
= Hen•ω(z)×∆−(He•)
(z) .
Consequently, we obtain:
Hen•ω(z + 1) = eωHen•ω(z) . (113)
Using the fact that the Borel transform induces an isomorphism fromC[[z−1]] to Cδ ⊕ Cζ, we can denote B(Hen•ω) by
B(Hen•ω)(ζ) = Hen•ωδ + Hen•ω(ζ) . (114)
38
Now, we can interpret (113) in the convolutive model, by taking its Boreltransform:
Hen•ωδ + e−ζHen•ω(ζ) = eωHen•ωδ + eωHen•ω(ζ) , (115)
which is equivalent to: (eω − 1)Hen•ω = 0• .
(eω − e−ζ)Hen•ω = 0• .(116)
Consequently, we have necessarily Hen•ω = 0• for all ω ∈ C, and hence
B(Hen•ω)(ζ) = Hen•ωδ , i.e. Hen•ω = Hen•ω ∈ C . (117)
Moreover, from (116), we deduce that Hen•ω = 0• if ω 6∈ 2iπZ .
To conclude the proof, we now have to prove that the mould Hen•ω isalternel , but this is a consequence of Property 2.
Equation (111), expressed in the convolutive model, gives for all n ∈ Z∗:
∆∆2inπHe∅
= 0 =⇒ Hen∅2inπ = 0 . (118)
∆∆2inπHes
= J(2inπ) δ =⇒ Hens2inπ = J(2inπ) . (119)
Nevertheless, there is, a priori, no easy way to compute the numbersHens1,··· ,sr2inπ if r ≥ 2 because it appears to be the residue at 2inπ of a com-
plicated combination of terms Heωωω
:
∆∆2inπHes1,··· ,sr
(ζ) = Hens1,··· ,sr2inπ δ +
r−1∑i=1
Hens1,··· ,si2inπ Hesi+1,··· ,sr
(ζ) . (120)
4.5. Application to the asymptotic expansion of Hurwitz multizeta functions
As an application of the resurgent character of Hurwitz multiple zetafunctions, and more precisely of Figure 4.1.3, we shall give the asymptoticexpansion of Hurwitz multizeta functions near infinity.
As a first calculation, we have
Lemma 14. Let k and r be two positive integers and s ≥ 2.
If A(ζ) =∑
n1,··· ,nr≥0
an1,··· ,nrζn1+···+nr+k
(n1 + · · ·+ nr + k)!and B(ζ) =
ζs−1
(s− 1)!are
two formal power series at the origin, then:
(A ? B) (ζ)
eζ − 1=∑
n1,··· ,nr+1≥0
(||n||+ k + s− 1
nr+1 − 1
)an1,··· ,nr
bnr+1
nr+1
ζ ||n||+k+s−1
(||n||+ k + s− 1)!,
39
where ||n|| = n1 + · · ·+ nr+1, bk denotes the k-th Bernoulli number and thesummand is interpreted to be equal to
an1,··· ,nrζ ||n||+k+s−1
(||n||+ k + s)!,
when nr+1 = 0.
Proof. This is a direct computation since:
(A ? B)(ζ) =∑
n1,··· ,nr≥0
an1,··· ,nrζn1+···+nr+k+s
(n1 + · · ·+ nr + k + s)!. (121)
Therefore, we obtain:
(A ? B)(ζ)
eζ − 1=
∑n1,··· ,nr+1≥0
an1,··· ,nrbnr+1
nr+1!
ζn1+···+nr+1+k+s−1
(n1 + · · ·+ nr + k + s)!. (122)
From
Hes1
(ζ) =∑n1≥0
(n1 + s1 − 2
s1 − 1
)bn1
n1
ζn1+s1−2
(n1 + s1 − 2)!(123)
for all integers s1 ≥ 2, we deduce from Lemma 14 that
Hes1,s2
(ζ) =∑
n1,n2≥0
(n1 + s1 − 2
n1 − 1
)(n12 + s12 − 3
n2 − 1
)bn1
n1
bn2
n2
ζn12+s12−3
(n12 + s12 − 3)!
(124)
and more generally
Hes1,··· ,sr
(ζ) =∑
n1,··· ,nr≥0
(r∏
k=1
(n1···k + s1···k − k − 1
nk − 1
)bnknk
)ζn1···r+s1···r−r−1
(n1···r + s1···r − r − 1)!,
(125)where n1···k denotes n1 + · · ·+ nk and s1···k = s1 + · · · sk .
Now, it is easy to deduce from it
Proposition 9. For any sequence (s1, · · · , sr) ∈ S?, Hes1,··· ,sr+ has an asymp-totic expansion near infinity given by
Hes1,··· ,sr
(z) =∑
n1,··· ,nr≥0
(r∏
k=1
(n1···k + s1···k − k − 1
nk − 1
)bnknk
)1
zn1···r+s1···r−r .
(126)
40
Let us notice that another proof of this asymptotic expansion could beobtained from a recursive use of the Euler-Maclaurin formula.
Example 6.
He2(z) =
1
z− 1
2z2+
1
6z3+ · · · (127)
He3(z) =
1
2z2− 1
z3+
1
4z4+ · · · (128)
He2,1
(z) =1
z− 3
2z2+
11
9z3+ · · · (129)
5. On the algebraic structure of Hmzvcv
In this section, we are going to study the algebraic structure of Hmzvcv,which is the algebra spanned by the convergent Hurwitz multizeta functions.The first question is the linear independence over C of the Hurwitz multiplezeta values. This has already been established in [33], p. 381.
As an easy consequence of Example 6, we can show that:
Lemma 15. The functions 1, He2+, He2,1
+ and He3+ are C-linearly indepen-
dant.
This method can be applied for any particular case, but will not be soeasy in a general way. So, we will go back to Lemma 4 to use anothermethod.
5.1. Linear independence of Hurwitz multizeta functions on the rationalfraction fields
Despite the fact we are going to use the C-linear independence in orderto study the algebraic structure of Hmzvcv, we will show more than this:
Theorem 3. The family(Hes+
)s∈S? is C(z)-linearly independent.
Let us notice that this result has not been pointed out in [33], even ifthe method is actually similar.
5.1.1. Preliminary examples
As a first example of the method developped in this section, let us con-sider three rational functions F2, F2,1 and G valued in C such that
F2He2+ + F2,1He2,1
+ = G . (130)
If F2,1 6= 0, we can assume that F2,1 = 1. Then, applying the operator ∆−to this relation, we obtain:
F2He2+ = G , (131)
41
where F2(z) = ∆−(F2)(z) +
1
z= ∆−(F2 +He1
+)(z)
G(z) = ∆−(G)(z)− 1
z2· F2(z − 1)
(132)
and
He1+(z) =
∑n>0
(1
n+ z− 1
n
). (133)
If F2 6= 0, Equation (131) would imply that He2+ is a rational fraction,
and therefore has a finite number of poles. But, each k ∈ N∗ is a pole ofHe2
+. Consequently, Equation (131) imposes F2 = G = 0 .Consequently, F2 +He1
+ is a 1-periodic function. Note that the rationalfraction F2 has a finite number of poles, so there exist N ∈ N∗ such that Nand −N are not a pole of F2, but N is a pole of He1
+. Consequently, −N isa pole of F2 +He1
+. Using the 1-periodicity, we therefore obtain that eachinteger is a pole of F2 +He1
+. This is, in particular, the case of the positiveinteger N , which show us a contradiction.
Therefore, F2,1 = 0, which also implies, as we have just seen from Equa-tion (131), that F2 = G = 0.
Thus, we can state the following
Lemma 16. The functions 1, He2+ and He2,1
+ are C(z)-linearly indepen-dent.
The following example will explain how an induction process can be usedto prove Theorem 3.
So, as a second example, let us consider this time rational functions F2,F2,1, F3 and G valued in C such that
F2He2+ + F3He3
+ + F2,1He2,1+ = G . (134)
We will apply exactly the same steps as in the previous example, that is:
1. assume that F3 6= 0 .
(a) assume more precisely that F3 = 1 .(b) apply the operator ∆− to Equation (134) to obtain
F2(z)He2+ + F2,1He2,1
+ = G(z) . (135)
where F2(z) = ∆−(F2)(z) +
1
z· F2,1(z − 1)
F2,1(z) = ∆−(F2,1)(z)
G(z) = ∆−(G)(z)− 1
z3− 1
z2· F2(z − 1)
(136)
42
(c) use Lemma 16 to conclude that:∆−(F2)(z) +
1
z· F2,1(z − 1) = 0 .
∆−(F2,1)(z) = 0 .
∆−(G)(z)− 1
z3− 1
z2· F2(z − 1) = 0 .
(137)
(d) partially solve System (137):F2,1 is a constant f2,1 ∈ C .
f2,1 · He1+ + F2 = 1− periodic function .
(138)
(e) obtain a contradiction in the same way as in the previous exam-ple, and thus F3 = 0.
2. rewrite Equation (134) as (130), and use Lemma 16 to conclude that
F2 = F2,1 = F3 = G = 0 . (139)
Finally, Lemmas 15 and 16 can be improved to:
Lemma 17. The functions 1, He2+, He2,1
+ and He3+ are C(z)-linearly inde-
pendent.
5.1.2. Proof of Theorem 3
Using the algorithm described in the second example of the previoussection, we will prove this theorem by an induction process based on thedegree of Hurwitz multizeta functions. But, before beginning the proof, letus introduce some notations, order relations and propositions.1. For all d ∈ N, let S?≤d and S?d be the sets defined by
S?≤d = s ∈ S? ; d˚s ≤ d ,
S?d = s ∈ S? ; d˚s = d ,(140)
where the degree of a sequence s is defined to be the difference between theweight and the length of the sequence s : d˚s = ||s|| − l(s).2. We can order the sequences of S?d+1 by numbering first the sequencesof length 1, then those of length 2, etc. For the following proof, let usremark that it will not be necessary to precise the numbering within a givenlength. So, we will consider the sets S?d+1 = sn ; n ∈ N∗ and, for n ∈ N,
S(d+1)n = si ; 1 ≤ i ≤ n . Of course, the sequences sn depend on d, but
we omit it for simplicity.3. We will finally consider the following properties:
D(d) : “the family(Hes+
)s∈S?≤d
is C(z)-free.”
P(d, n) : “the family(Hes+
)s∈S?≤d
⋃(Hes+
)s∈S(d+1)
nis C(z)-free.”
(141)
43
Proving the theorem amounts to proving that the property D(d) is truefor all d ∈ N. Since S?≤0 = ∅, the property D(0) is obviously true. Conse-quently, we have to prove the heredity of D(n) .
This will also be done by an induction process, showing that we canadd one by one the sequences of S?d+1 in the statement given by D(n) toobtain the property D(d+ 1): the heredity of P(d, n) will justify this. SinceP(d, 0) = D(d), the initialisation of the property P(d, n) is obvious.
In other terms, we will prove the property P(d, n) by induction on(d, n) ∈ N2, where N2 is ordered by the lexicographic order.
Then, the proof of the theorem boils down to showing the followingimplication:
∀(d, n) ∈ N2 , P(d, n) =⇒ P(d, n+ 1) ,
which is what we will now do.
Proof. Assume Property P(d, n) for a pair (d, n) ∈ N2 and let us showthat P(d, n+ 1) is, therefore, true .
In order to do this, let us consider the relation:∑s∈S?≤d∪S
(d+1)n+1
s6=∅
FsHes+ = F , (142)
where F and Fs , s ∈ S?≤d ∪ S(d+1)n+1 , are rational fractions.
We will show that Fsn+1 = 0 by exhibiting a contradiction. Consequently,the property P(d, n) will imply that all others rational fractions are zero ;therefore, the property P(d, n + 1) will be proven. So, let us now assumethat Fsn+1 6= 0 .
Up to dividing (142) by Fsn+1 , we can assume without loss of generalitythat Fsn+1 = 1 .
Let us set
sn+1 = u · p with
p ≥ 1 .u ∈ S?≤d+2−p .
(143)
Let us remark that it is possible to have u = ∅ when n = 0 . Moreover, wewill use the notation Fs even if the sequences s do not appear a priori inRelation (142) ; in this case, we will set Fs = 0 .
Step 1 : Application of ∆− on Relation (142) .
44
Since ∆− is a (1 ; τ−1)-derivative, if we apply it to Relation (142), weobtain:
∆−
(Hes
n+1
+
)+
∑s∈S?≤d∪S
(d+1)n
s 6=∅
(∆−
(Fs
)·Hes++τ−1
(Fs
)·∆−
(Hes+
) )= ∆−(F ) .
(144)Using a change of index, we obtain:∑
s∈S?≤d∪S(d+1)n
s6=∅
τ−1(Fs
)·∆−
(Hes+
)=
∑s∈S?≤d∪S
(d+1)n
s 6=∅
τ−1(Fs
)· Hes
<r
+ · Jsr
=∑
s∈S?≤d∪S(d+1)n
∑k∈N∗
s·k∈S?≤d∪S(d+1)n
τ−1(Fs·k
)· Jk · Hes+ .
(145)So, we have:
∆−(F ) = Jp · Heu+ +∑
s∈S?≤d∪S(d+1)n
s 6=∅
∆−(Fs
)· Hes+
+∑
s∈S?≤d∪S(d+1)n
∑k∈N∗
s·k∈S?≤d∪S(d+1)n
τ−1(Fs·k
)· Jk · Hes+ .
(146)
The Hurwitz multizeta Hes+ that appear in this relation all satisfy s ∈S?≤d ∪ S
(d+1)n . Indeed, let us recall that we have numbered the sequences of
the set S?d+1 by the length. Since ∆−
(Hes
n+1
+
)= ∆−
(Heu·p+
)= Heu+ · Jp ,
we therefore have u ∈ S?≤d ∪ S(d+1)n .
We obtain the following system by distinguishing, in the third term ofthe last equation, whether the summation sequence s is empty or not, andthen by an application of the induction hypothesis P(d, n):
∀s ∈(S?≤d ∪ S
(d+1)n
)− ∅ , ∆−(Fs) +
∑k∈N∗
s·k∈S?≤d∪S(d+1)n
τ−1(Fs·k) · Jk + δu,sJp = 0 .
∆−(F ) =d+1∑k=2
τ−1(Fk)Jk + (1− δn,0)τ−1(Fd+2)Jd+2 .
(147)
Step 2: A lemma which gives some partial solutions to System (147) .
45
The contradiction will be highlighted by the following lemma. So, wewill look for a relation as in it, which will be found by a partial solving ofthe previous system and will be done in Lemma 19.
Lemma 18. Let F be a rational fraction and f a 1-periodic function.If, for an n-tuple (λ1, · · · , λn) ∈ Cn , with n ∈ N, the equality
F +
n∑i=1
λiHei+ = f (148)
is valid, then we necessarily have:
λ1 = · · · = λn = 0 .F and f are constant functions .
Proof. Let us assume thatn∑i=1
λiHei+ 6= 0 . Then, this function has its
poles located at negative integers.Besides, there exists N ∈ N such that N and −N are not poles of F . Thus,−N is a pole of f , which therefore admits all integers as poles, according toits 1-periodicity. Since N is a pole of f , N must be a pole of either F , or ofn∑i=1
λiHei+ 6= 0 , which highlights a contradiction.
Then, we have:
1.
n∑i=1
λiHei+ = 0 , which produces λ1 = · · · = λn = 0 . In fact, when
applying ∆−, we obtain:n∑i=1
λiJi = 0, which gives λ1 = · · · = λn = 0
according to the partial fraction expansion uniqueness.
2. F = f , which proves that F and f are constant functions.
According to this lemma, we will now be able to solve partially System
(147). Let us remind that we have noted sn+1 = u·p, where
p ≥ 1 .u ∈ S?≤d+2−p .
Lemma 19. Let r be a positive integer and p ≥ 2 .Let us also consider two r-tuples, (n1; · · · ;nr) ∈ Nr and (k1; · · · ; kr) ∈ (N∗)r
such thatr∑i=1
(ki − 1) ≤ p− 2 .
Then, Fu·k1·1[n1]····kr·1[nr ] =
0 , if nr > 0 .
cste , if nr = 0 .
46
Proof. Let us denote v = u·k1 ·1[n1] · · ··kr and δ(v) = p− 2−r∑i=1
(ki − 1) .
Let us show this lemma by induction on δ(v) (even if δ(v) can only take afinite number of values).
• If δ(v) = 0, System (147) applied to v · 1[n], n ∈ N, give us:
∀n ∈ N , ∆−
(Fv·1[n]
)+ τ−1
(Fv·1[n+1]
)J1 = 0 . (149)
In fact:
v · 1[n] · k ∈ S?≤d ∪ Sn ⇐⇒ d˚(v · 1[n] · k) ≤ d⇐⇒ d˚u− δ(v) + p+ k − 3 ≤ d⇐⇒ k ≤ δ(v) + 1⇐⇒ k = 1 (150)
We know that there exists n0 ∈ N such that Fv·1[n0] = 0 , that is to
say that the sequence v · 1[n0] does not appear in (142). Then, (149) gives
∆−
(Fv·1[n0−1]
)= 0 . Thus, Fv·1[n0−1] is a 1-periodic function, and therefore
a constant function according to the previous lemma, which will now be de-
noted by fv·1[n0−1] . (149) gives us again4: ∆−
(Fv·1[n0−2] + fv·1[n0−1]He1
+
)=
0, so Fv·1[n0−2] + fv·1[n−1]He1+ is a 1-periodic function. Again, the pre-
vious lemma imposes to Fv·1[n0−2] being a constant function as well asFv·1[n0−1] = fv·1[n0−1] being the null function.
We obtain, by repeating the same reasoning:
Fu·k1·1[n1]····kr·1[nr ] =
0 , if nr > 0 .
cste , if nr = 0 .(151)
• Let us assume the lemma proved for all sequence v such that δ(v) ≤ kand let us show it for sequences such that δ(v) = k + 1 .
System (147) applied this time to v · 1[n], n ∈ N, gives:
∆−
(Fv·1[n]
)+ τ−1
(Fv·1[n+1]
)J1 +
δ(v)+1∑l=2
τ−1(Fv·1[n]·l
)J l = 0 . (152)
According to the induction hypothesis, we have that for all l ∈ [[ 2 ; δ(v)+1 ]] , Fv·1[n]·l is a constant function which will be, as usual, denoted by fv·1[n]·l .
4Let us remark that we have ∆−(He1+) = J1, but this does not come from Prop-
erty 4 because we have, in it, ruled out the case of divergent Hurwitz multizeta functions.Nevertheless, this equality is true.
47
Consequently, the last equation can be written as:
∀n ∈ N , ∆−
(Fv·1[n]
)+ τ−1
(Fv·1[n+1]
)J1 +
δ(v)+1∑l=2
fv·1[n]·lJl = 0 . (153)
We will now apply the same reasoning as in the case δ(v) = 0. The onlymodification we have to do is the application of the previous lemma to therelation
Fv·1[n−1] +
δ(v)+1∑l=1
fv·1[n−1]·lHel+ = 0 (154)
instead of Fv·1[n−1] + Fv·1[n]He1+ = 0 .
So, we have again proved that:
Fu·k1·1[n1]····kr·1[nr ] =
0 , si nr > 0 .
cste , si nr = 0 .(155)
• The lemma is thus proved for any value of δ(v) .
Step 3 : Revealing of the contradiction.
Equation (147), applied to u, give us:
∆−(Fu) +
p−1∑k=1
τ−1(Fu·k)Jk + Jp = 0 . (156)
The previous lemma gives us, in particular, that Fu·1 , · · · , Fu·p−1 aresome rational fractions which are constant functions and now denoted byfu·1 , · · · , fu·p−1. This can be rewritten:
∆−
(Fu +
p−1∑k=1
fu·kHek+ +Hep+
)= 0 . (157)
Thus, Fu +
p−1∑k=1
fu·kHek+ +Hep+ defines a 1-periodic function. The coeffi-
cients being not all zero in this relation, this enlights a contradiction ofLemma 18 .
Thus, Fsn+1 = 0 and (142) can be rewritten:
F =∑
s∈S?≤d∪Sn
FsHes+ . (158)
48
It only remains to use the induction hypothesis P(d, n) to obtain thatall other rational fractions are null. Therefore, we have shown that:
P(d, n) =⇒ P(d, n+ 1) .
This completes the proof of the theorem.
5.2. Algebraic relations in HmzvcvIn this section, we give some corollaries of Theorem 3. We will begin
with some reminders on the algebra QSym of quasi-symmetric functions, andthen we will give an explicit isomorphism between this algebra and Hmzfcv.Thus, we will be able to raise up any relation between convergent Hurwitzmultizeta functions in QSym. Consequently, we will give an answer, for thealgebra of convergent Hurwitz multizeta functions, to each question we askand would be able to answer for the algebra of multizeta values.
5.2.1. Reminders on QSym
Let us consider an infinite commutative alphabet X = x1, x2, x3, · · · .Thus, we can consider (commutative) formal power series, valued in a ringR, with indeterminates x1, x2, · · · . In particular, we are interested in seriessuch that the coefficient of the monomial xs11 · · ·xsrr is equal to the coefficientof the monomial xs1n1
· · ·xsrnr for any strictly increasing sequence 0 < n1 <n2 < · · · < nr of positive integers indexing the variables (and for any positiveinteger sequence of exponents s1, · · · , sr).
Such a series is called a quasi-symmetric function(see [28], or [3], [30],[31] and [39] for a more recent presentation).
From the definition, it is clear that the set of quasi-symmetric functionsis an R-vector space denoted QSymR(X), or more simply QSymR, as wellas QSym if there is no possible ambiguity.
This vector space has a natural basis, the so-called monomial basis. Letus notice that QSym has other interesting bases (see [3], for the fundamen-tal basis, but also [47] for a new basis related to multizetas values) . Acomposition I of length r being given, we denote by
MI(X) =∑
0<n1<···<nr
xi1n1· · ·xirnr , (159)
the monomial basis. These series are quasi-symmetric functions. Moreover,according to the definition, it turns out that they clearly span QSym. Onthe other hand, if we order the indeterminates by
x1 > x2 > x3 > · · · , (160)
and extend this order to words to be the lexicographic order, then the leadingterm of MI(X) is xi11 · · ·xirr . Thus, the MI ’s are also linearly independentand consequently are a basis of QSym.
49
Since the monomials MI are iterated sums, the MI ’s multiply themselvesusing the stuffle product (see [32], for instance). Therefore, the product oftwo quasi-symmetric functions is a quasi-symmetric function, and QSym isactually an R-algebra.
Let us conclude these reminders by noticing that we have given theclassical definition of quasi-symmetric function: the sequence of positiveintegers indexing the variables is a strictly increasing sequence. Nevertheless,the multizetas’ convention, which is the most used today, and consequentlythe Hurwitz multizetas’ convention, is the opposite one: the sequence ofpositive integers indexing the variables is a strictly decreasing sequence.
If the both conventions were identically the same, the Hurwitz multizetafunctions should be seen as a particular evaluation of the monomials via the
specialisation xn 7−→1
n+ z, according to the condition that the result is
a convergent series. In order to use jointly both of these conventions, wecan consider the alphabet X = x−1;x−2;x−3; · · · and the specialisation
x−n 7−→1
n+ z. However, to make links between QSym and multizeta
values, we only need to have an infinite totally ordered alphabet.
5.2.2. Corollaries to Theorem 3
A totally ordered and infinite alphabet A = a1, a2, · · · being given, aword ω = w1w2 · · ·wl over A is said to be a Lyndon word if it is strictlysmaller than any of its non-empty proper right factors for the lexicographicorder:
ω < wiwi+1 · · ·wl for all i ∈ [[ 2 ; l ]] . (161)
To have more information about Lyndon words, we refer the reader to[37] for historical references as well as to [50].
The definition of Lyndon words allows us to deduce from Theorem 3 aring basis of Hmzfcv:
Corollary 2. Let us denote by QSymcv the subalgebra of QSym spannedby the monomials MI with a composition I = (i1, · · · , ir) such that i1 ≥ 2 .Then:
Hmzfcv ' QSymcv ' Q[Lyn(N∗)− 1
], (162)
where Lyn(y1; y2; · · · ) denotes the set of Lyndon words over the alphabetY = y1; y2; · · · .
Proof. 1. Let us define h : QSymcv −→ Hmzfcv on the monomial basisby h(Ms1,··· ,sr) = Hes1,··· ,sr+ .h is a algebra morphism, since the monomial basis of QSym and the Hurwitzmultizeta functions are multiplied by the same rule, the stuffle product.
50
Moreover, h is a surjective map, by construction. Finally, h is injective,according to Theorem 3. Consequently, h is an algebra isomorphism.
2. A stuffle contains all the terms of the ordinary shuffle together withcontractions obtained by adding two consecutive parts coming from differentterms. So, by a triangular change of variable, the quasi-shuffle productdefines an algebraic structure, which is actually isomorphic to an ordinaryshuffle algebra over the same set (see [32]).
According to the Radford theorem (see [48]), a shuffle algebra over thealphabet A is a polynomial algebra in the Lyndon words over A as genera-tors. Thus, a stuffle algebra over the alphabet A is also a polynomial algebraspanned by an algebraically independent familly indexed by Lyndon wordsover A (see [32]). Consequently, if we define a subalgebra of a stuffle algebra,we just have to exclude some relevant Lyndon words:
QSymcv ' Q⟨Lyn(y1; y2; · · · )− y1
⟩(163)
Since it is well-known that QSym is a graded algebra, QSymcv is also agraded algebra whose nth homogeneous component is of dimension 2n−2 (andrespectively 1 and 0 if n = 0 and n = 1). Consequently, the isomorphismintroduced in Corollary 2 first implies:
Corollary 3. Let us denote the algebra of convergent Hurwitz multizetafunctions of weight n by Hmzfcv,n, that is the subalgebra of Hmzfcv spannedby the Hurwitz multizetas Hes1,··· ,sr+ such that s1 + · · ·+ sr = n and s1 ≥ 2 .Then:
1. Hmzfcv is graded by the weight: Hmzfcv =⊕n∈NHmzfcv,n .
∀(p, q) ∈ N2 , Hmzfcv,p · Hmzfcv,q ⊂ Hmzfcv,p+q .(164)
2.
dim Hmzfcv,0 = 1 , dim Hmzfcv,1 = 0.
dim Hmzfcv,n = 2n−2 for all n ≥ 2.
Finally, the isomorphism h allows us to completely describe the alge-braic relations between convergent Hurwitz multizeta functions since theserelations can be lifted to the formal level:
Corollary 4. Each algebraic relation in Hmzfcv comes from the expansionof the stuffle products.
51
Proof. Let us consider the algebra A = Q〈Ω〉, where
Ω =⋃n∈NyI ; I composition of n , (165)
with the concatenation product. Let us recall that the stuffle product isrecursively defined by (17) for any alphabet which has a semi-group struc-ture. Thus, this definition is a valid one for the alphabet N1 and give us, bybilinearity, a stuffle over A:
yI yJ =∑K∈N?1
〈I J |K〉yK (166)
Here, it is natural to use the same symbol to indicate the stuffle over Aand over the compositions since it is morally the same product.
Let I be the ideal of A spanned by the elements yI ·yJ−yI yJ , where Iand J are two compositions and · denote the concatenation product. Thus,A/I is an algebra, which is naturally graded since I is an homogeneous idealand A is graded by the weight (with dyI = n, if I is a composition of n).
The homogeneous component of weight n of A/I is of dimension 2n−1,since each element of A/I is generated by the yI , I being a composition ofn, these elements are linearly independent (according to the definition of A)and are 2n−1.
Finally, let us consider the linear map ϕ : A/Y −→ QSym defined byϕ(yI) = MI . It is an algebra morphism:
ϕ(yI · yJ) = ϕ(yI yJ) = ϕ( ∑K∈N?1
〈I J |K〉yK)
=∑K∈N?1
〈I J |K〉MK = MIMJ = ϕ(yI)ϕ(yJ) .
(167)
Moreover, ϕ, restricted to the homogeneous component of weight n of A/I,is valued in QSymn, is surjective by definition and injective according to thedimensions. Consequently, ϕ is an isomorphism between A/Y and QSym.
If we have an algebraic relation between convergent Hurwitz multizetafunctions, like
n∑i=1
λi
(Hes
i1
+
)αi1 · · ·(Hesini+
)αini= 0 (168)
where, for all i ∈ [[ 1 ; n ]], λi ∈ C, ni ∈ N∗ and sij ∈ S?, we can lift it to theformal level, from Hmzfcv to QSymcv, and then to A/I:
n∑i=1
λi
(ysi1
)αi1 · · ·(ysini)αini = 0 in A/I . (169)
52
Consequently:
n∑i=1
λi
(ysi1
)αi1 · · ·(ysini)αini ∈ I . (170)
Therefore, all the algebraic relations between convergent Hurwitz multi-zeta functions, like (168), is directly a consequence of the stuffle relations.
6. Extension to the divergent Hurwitz multizeta functions
First of all, let us remind the following Lemma from [6] which gives usan algebraic way to extend the definition of a symmetrel mould:
Lemma 20. Let Se• be a symmetrel mould over the alphabet Ω = N∗, withvalues in a commuative algebra A, well-defined for sequences in S? = s ∈N?1 ; s1 ≥ 2
1. For all θ ∈ A , there exists a unique symmetrel extension of Se• toN?1, denoted by Se•θ , such that Se1
θ = θ .
2. For all γ ∈ A , let N e•γ be the symmetrel mould defined on sequences
of N?1 by: N esγ =
γr
r!, if s = 1[r] .
0 , otherwise.
Then, for all (θ1 ; θ2) ∈ A2 , we have:
Se•θ1 = N e•θ1−θ2 × Se•θ2 . (171)
6.1. Algebraic structure of HmzfAccording to Lemma 20, to extend the definition of the Hurwitz multi-
zeta functions to the divergent case, i.e. when s1 = 1, we just need to choosewhich function will be He1
+. Let us notice that, for any choice, the algebraHmzv spanned by all the Hurwitz multizeta functions (the convergent anddivergent ones) is then described by
Hmzf = Hmzfcv[He1
+
]. (172)
Thus, Theorem 3 can be extended to all Hurwitz multizeta functions:
Theorem 4. The family(Hes+
)s∈N?1
is C(z)-linearly independent.
Its corollary can also be extended easily to obtain:
Corollary 5. Hmzf ' QSym ' Q[Lyn(N∗)
], where Lyn(y1; y2; · · · ) de-
notes the set of Lyndon words over the alphabet Y = y1; y2; · · · .
53
Corollary 6. Let us denote the algebra of Hurwitz multizeta functions ofweight n by Hmzfn .Then:
1. Hmzf is graded by the weight: Hmzf =⊕n∈NHmzfn .
2.
dim Hmzf0 = 1 ,
dim Hmzfn = 2n−1 for all n ≥ 1.
Corollary 7. Each algebraic relation in Hmzv comes from the expansionof the stuffle products.
6.2. About a possible resurgent character of divergent Hurwitz multizetafunctions
Since the properties of convergent Hurwitz multizeta are to be extended,we want to define the divergent Hurwitz multizeta He1
+ in order that itsatisfies:
1. He1+ is defined from a Borel-Laplace summation of a certain f ∈
− ln z + C[[z−1]] satisfying the difference equation
∆−f(z) =1
z. (173)
2. He1+ is asymptotically equal, near infinity, to − ln z +
∑n>0
bnnzn
, where
bn is the nth Bernoulli number.
If there exists such a function He1+, this one is unique. Indeed, the first
point gives us that ∆−f′(z) =
−1
z2with f ′ ∈ C[[z−1]]. Thus, we necessarily
have f fixed, up to a constant since f ′ = −He2. Then, the second point
prove the unicity of the constant. Moreover, provided such a function exists,this last differential property can be generalized to all divergent Hurwitzmultizeta functions such that Equation (5) is valid for any divergent Hurwitzmultizeta functions.
On the other hand, it is clear that (173) has a unique solution f(z) =− ln z + g(z) ∈ − ln z + C[[z−1]] caracterized by:
∆−g(z) =1
z+ ln
(1− 1
z
)∈ z−2C[[z−1]] . (174)
Then, the resurgent treatment of the generic 1-order difference equation canbe used to produce a simple resurgent function defined in the geometric
54
model for all z ∈ C, <e z > 0, by
ge(z) =∑n>0
[1
n+ z+ ln
(1− 1
n+ z
)]= ln z + γ +
∑n>0
(1
n+ z− 1
n
)= ln z − 1
z− Γ′(z)
Γ(z), (175)
where γ ' 0.5772156649 denotes the Euler - Mascheroni constant and Γ theGamma function which satisfies the well-known identity:
Γ(z + 1) = zΓ(z) (176)
Consequently, the function f defined over <e z > 0 by
f(z) = −1
z− Γ′(z)
Γ(z)= γ +
∑n>0
(1
n+ z− 1
n
)(177)
satisfies as required (173) and thus the first point.
Finally, it is well-known that the derivative of the logarithm of the Γfunction has an asymptotic expansion near infinity given by
Γ′(z)
Γ(z) ln z − 1
2z−∑n>0
b2n2nz2n
. (178)
which shows that f also satisfies the second point. Thus, we shall choose
He1+(z) = γ +
∑n>0
(1
n+ z− 1
n
)(179)
which allows us to extend multizeta values to the divergent case by settingZe1 = γ, which is exactly the choice one would have, according to [55]. Butfor simplicity, we prefer to choose Ze1 = 0 and use, if necessary, the “changeof constant formula” (171). Therefore, we set:
He1+(z) =
∑n>0
(1
n+ z− 1
n
)(180)
Let us notice that this regularization is actually set out in [55] for thedivergent multizeta values from a different point of view, namely this ofspecial functions. It is nice to see that the resurgent point of view agreeswith this one.
From the previous resurgent treatment of (173), we now know that He1+
is equal to a simple resurgent function minus the principal branch of thecomplex logarithm. Therefore, Theorem 1 is extended as:
55
Theorem 5. Any Hurwitz multizeta function is a polynomial in − ln withcoefficients in the algebra of simple resurgent functions whose singularitiesare over 2πiZ∗ .
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